Surds

Surds are irrational numbers that cannot be expressed as exact decimals or fractions. They are usually represented as square roots of non-perfect square numbers.

When working with surds, it’s important to simplify expressions by factoring out perfect squares and rationalising denominators when necessary. By doing this, surds can be presented in a simpler and more manageable form, making them easier to work with in mathematical problems.

Let’s look at how to simplify and rationalise surds.

Simplifying Surds

To simplify a surd, we use the rule: $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ .

Consider the following examples:

$\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$

$\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$

Now, let’s simplify an expression involving surds:

$\sqrt{12} + \sqrt{300} - \sqrt{48} = \sqrt{4 \times 3} + \sqrt{100 \times 3} - \sqrt{16 \times 3}$

$= 2\sqrt{3} + 10\sqrt{3} - 4\sqrt{3} = 8\sqrt{3}$

Ok, let’s look at some more examples.

Example 1:

Simplify $\frac{2}{\sqrt{2}}$

$\frac { 2 }{ \sqrt { 2 } } =\frac { 2 }{ \sqrt { 2 } } \times \frac { \sqrt { 2 } }{ \sqrt { 2 } }$

$=\frac { 2\sqrt { 2 } }{ 2 }$

$=\sqrt { 2 }$

Example 2:

Simplify $(\sqrt { 2 } +\sqrt { 5 } )^2$

Start by expanding the expression:

$(\sqrt { 2 } +\sqrt { 5 } )^2 = (\sqrt { 2 } +\sqrt { 5 } )(\sqrt { 2 } +\sqrt { 5 } )$

Now, multiply the terms:

$=2+\sqrt { 2 } +\sqrt { 5 } +\sqrt { 5 } +\sqrt { 2 } +5$

Combine like terms:

$=7+\sqrt { 10 } +\sqrt { 10 }$

The simplified expression is:

$= 7 + 2\sqrt { 10 }$

Example 3:

Simplify $(\sqrt { 3 } +\sqrt { 2 } )^2 - (\sqrt { 3 } -\sqrt { 2 } )^2$

We can use the difference of squares formula $a^2 - b^2 = (a + b)(a - b)$ with:

$a = \sqrt { 3 } +\sqrt { 2 }$ and $b = \sqrt { 3 } -\sqrt { 2 }$

Compute $a + b$ and $a - b$ :

$a + b = \sqrt { 3 } +\sqrt { 2 } +\sqrt { 3 } -\sqrt { 2 } = 2\sqrt { 3 }$

$a - b = \sqrt { 3 } +\sqrt { 2 } - (\sqrt { 3 } -\sqrt { 2 }) = 2\sqrt { 2 }$

Remember, $\sqrt { A } - \sqrt { A } = 0 and \sqrt { A } + \sqrt { A } = 2\sqrt { A }$

Now apply the difference of squares formula:

$(\sqrt { 3 } +\sqrt { 2 })^2 - (\sqrt { 3 } -\sqrt { 2 })^2 = (2\sqrt { 3 })(2\sqrt { 2 })$

Multiply the terms:

$= 4\sqrt { 6 }$

The simplified expression is:

$= 4\sqrt { 6 }$

Rationalising Surds

When working with surds (square roots of non-perfect squares), sometimes we need to rationalise the denominator, which means removing the surds from the denominator. Let’s look at how to rationalise fractions containing surds in their denominators.

Conjugates

The conjugate of $\sqrt{a} +\sqrt{b}$ is $\sqrt{a} -\sqrt{b}$ , and the conjugate of $\sqrt{a} -\sqrt{b}$ is $\sqrt{a} +\sqrt{b}$ . Conjugates are a useful tool for rationalising surds. To illustrate the process, let’s look at some examples.

Keep in mind this key formula when working with conjugates:

$(a+b)(a-b)=(a-b)(a+b)=a^{2}-b^{2}$

It’s also important to remember this formula:

$(a-b)(a-b)=a^{2}+b^{2}-2ab$

Example 1:

Rationalise $\frac{5}{\sqrt{3}}$

$\frac{5}{\sqrt{3}} = \frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$

Example 2:

Rationalise the following:

1. $\frac{8}{\sqrt{2}}$

2. $\frac{12}{\sqrt{3}}$

1. $\frac{8}{\sqrt{2}} = \frac{8}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{8\sqrt{2}}{2}$

$= 4\sqrt{2}$

2. $\frac{12}{\sqrt{3}} = \frac{12}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{12\sqrt{3}}{3}$

$= 4\sqrt{3}$

Example 3:

Rationalise $\dfrac{1}{\left( \sqrt{2}-1\right) }$

To rationalise this fraction, multiply the numerator and denominator by the conjugate of $\sqrt{2}-1$ , which is $\sqrt{2}+1$ :

$\dfrac{1}{\sqrt{2}-1}=\dfrac{1}{\sqrt{2}-1}\times \dfrac{\sqrt{2}+1}{\sqrt{2}+1}$

Now apply the formula $(a+b)(a-b)=a^{2}-b^{2}$ :

$=\dfrac{\sqrt{2}+1}{(\sqrt{2})^{2}-(1)^{2}}$

Simplify the expression:

$=\dfrac{\sqrt{2}+1}{2-1}$

$=\sqrt{2}+1$

Example 4:

Rationalise: $\frac { 1 }{ \sqrt { 2 } +1 }$

To rationalise this fraction, multiply the numerator and denominator by the conjugate of $\sqrt { 2 } +1$ , which is $\sqrt { 2 } -1$ :

$\frac { 1 }{ \sqrt { 2 } +1 } =\frac { 1 }{ \sqrt { 2 } +1 } \times \frac { \sqrt { 2 } -1 }{ \sqrt { 2 } -1 }$

Now apply the formula $(a+b)(a-b)=a^{2}-b^{2}$ :

$=\frac { \sqrt { 2 } -1 }{ (\sqrt { 2 })^{ 2 } -(1)^{ 2 } }$

Simplify the expression:

$=\frac { \sqrt { 2 } -1 }{ 2-1 }$

$=\sqrt{2}-1$

Example 5:

Rationalise $\frac { 1+\sqrt { 2 } }{ 1-\sqrt { 2 } }$

To rationalise the denominator, multiply the numerator and denominator by the conjugate of the denominator.

$\frac { 1+\sqrt { 2 } }{ 1-\sqrt { 2 } } =\frac { 1+\sqrt { 2 } }{ 1-\sqrt { 2 } } \times \frac { 1+\sqrt { 2 } }{ 1+\sqrt { 2 } }$

This results in:

$=\frac { (1+\sqrt { 2 } )^2 }{ 1^2-(\sqrt { 2 })^2 }$

$=\frac { (1+\sqrt { 2 } ) (1+\sqrt { 2 } )}{ 1-2 }$

$=\frac { 1+2\sqrt { 2 } +2 }{ -1 }$

$=-(3+2\sqrt { 2 } )$

Simplified, we get:

$=-3-2\sqrt { 2 }$

Example 6:

Rationalise $\frac { 2-\sqrt { 3 } }{ \sqrt { 2 } -1 }$

Multiply by the conjugate of the denominator:

$\frac { 2-\sqrt { 3 } }{ \sqrt { 2 } -1 } =\frac { 2-\sqrt { 3 } }{ \sqrt {2 } -1 } \times \frac { \sqrt { 2 } +1 }{ \sqrt { 2 } +1 }$

This results in:

$=\frac { (2-\sqrt { 3 })(\sqrt { 2 } +1) }{ (\sqrt { 2 } -1)(\sqrt { 2 } +1) }$

$=\frac { 2\sqrt { 2 } +2-\sqrt { 6 } -\sqrt { 3 } }{ 2-1 }$

Simplified, we get:

$=2\sqrt { 2 } +2-\sqrt { 6 } -\sqrt { 3 }$

Example 7:

Rationalise $\frac { 27 }{ 9-\sqrt { 3 } }$

Multiply by the conjugate of the denominator:

$\frac { 27 }{ 9-\sqrt { 3 } } =\frac { 27 }{ 9-\sqrt { 3 } } \times \frac { 9+\sqrt { 3 } }{ 9+\sqrt { 3 } }$

This results in:

$=\frac { 27(9+\sqrt { 3 }) }{ 9^2-(\sqrt { 3 })^2 }$

$=\frac { 27(9+\sqrt { 3 }) }{ 81-3 }$

$=\frac { 27(9+\sqrt { 3 }) }{ 78 }$

Simplify the expression:

$=\frac { 27\times9 + 27\times\sqrt { 3 } }{ 78 }$

$=\frac { 243 + 27\sqrt { 3 } }{ 78 }$

Finally, we can simplify further by dividing both the numerator and the denominator by their highest common factor, which is 3:

$=\frac { 81 + 9\sqrt { 3 } }{ 26 }$

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Last updated: April 15, 2022

This Privacy Policy describes Our policies and procedures on the collection, use and disclosure of Your information when You use our services. It also tells you about your privacy rights and how the law protects you.

We use Your Personal data to provide and improve our services. By using our services, you are agreeing to the collection and use of information in accordance with this Privacy Policy.

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Payments

We provide paid services on our website and we may provide paid products. So, we may use third-party services for payment processing (e.g. payment processors).

We will not store or collect your payment card details. That information is provided directly to our third-party payment processors whose use of your personal information is governed by their privacy policy. These payment processors adhere to the standards set by PCI-DSS as managed by the PCI Security Standards Council, which is a joint effort of brands like Visa, Mastercard, American Express and Discover. PCI-DSS requirements help ensure the secure handling of payment information. We are currently using:

• Stripe
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Behavioural Remarketing

We use remarketing services to advertise to you after you have accessed or visited our website. We and our third-party vendors use cookies and non-cookie technologies to help us recognize your device and understand how you use our website. Therefore, we can improve our services to reflect your interests and serve you advertisements that are likely to be of more interest to you.

These third-party vendors collect, store, use, process and transfer information about your activity on our website in accordance with their Privacy Policies, which enables us to:

Measure and analyse traffic and browsing activity on our website
Show advertisements for our products and/or services to you on third-party websites or apps
Measure and analyse the performance of our advertising campaigns
Some of these third-party vendors may use non-cookie technologies that may not be impacted by browser settings that block cookies. Your browser may not permit you to block such technologies. Instead, you can use the following third-party tools to decline the collection and use of information for the purpose of serving you interest-based advertising:

The NAI’s opt-out platform: https://www.networkadvertising.org/choices/
The EDAA’s opt-out platform https://www.youronlinechoices.com/
The DAA’s opt-out platform: https://optout.aboutads.info/?c=2&lang=EN
You may opt-out of all personalised advertising by enabling privacy features on your mobile device, such as Limit Ad Tracking (iOS) and Opt Out of Ads Personalisation (Android). Check your mobile device Help system for more information.

We may share information, such as hashed email addresses (if available) or other online identifiers collected on our website with these third-party vendors. This allows our third-party vendors to recognise and deliver you ads across devices and browsers. To read more about the technologies used by these third-party vendors and their cross-device capabilities please refer to the privacy policy of each vendor listed below.

The third-party vendors We use are:

• Google Ads (AdWords)
Google Ads (AdWords) remarketing service is provided by Google Inc.

You can opt-out of Google Analytics for display advertising and customise the Google Display Network ads by visiting the Google Ads Settings page: https://www.google.com/settings/ads

Google also recommends installing the Google Analytics Opt-out Browser Add-on – https://tools.google.com/dlpage/gaoptout – for your web browser. The Google Analytics Opt-out Browser Add-on provides visitors with the ability to prevent their data from being collected and used by Google Analytics.

For more information on the privacy practices of Google, please visit the Google Privacy & Terms web page: https://policies.google.com/privacy

• Facebook
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You can learn more about interest-based advertising from Facebook by visiting this page: https://www.facebook.com/help/516147308587266

To opt-out from Facebook’s interest-based ads, follow these instructions from Facebook: https://www.facebook.com/help/568137493302217

Facebook adheres to the Self-Regulatory Principles for Online Behavioural Advertising established by the Digital Advertising Alliance. You can also opt-out from Facebook and other participating companies through the Digital Advertising Alliance in the USA https://www.aboutads.info/choices/, the Digital Advertising Alliance of Canada in Canada https://youradchoices.ca/ or the European Interactive Digital Advertising Alliance in Europe https://www.youronlinechoices.eu/, or opt-out using your mobile device settings.

For more information on the privacy practices of Facebook, please visit Facebook’s Data Policy: https://www.facebook.com/privacy/explanation

Usage, Performance and Miscellaneous

We may use third-party Service Providers to provide a better improvement of our website. For example, we use:

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We use an invisible captcha service named reCAPTCHA. which is operated by Google.

The reCAPTCHA service may collect information from you and from your device for security purposes.

The information gathered by reCAPTCHA is held in accordance with the Privacy Policy of Google: https://www.google.com/intl/en/policies/privacy/

GDPR Privacy

Legal Basis for Processing Personal Data under GDPR

We may process personal data under the following conditions:

• Consent: You have given Your consent for processing personal data for one or more specific purposes.

• Performance of a contract: The provision of personal data is necessary for the performance of an agreement with you or for any pre-contractual obligations thereof.

• Legal obligations: Processing personal data is necessary for compliance with a legal obligation to which the company is subject.

• Vital interests: Processing personal data is necessary in order to protect your vital interests or of another natural person.

• Public interests: Processing personal data is related to a task that is carried out in the public interest or in the exercise of official authority vested in the Company.

• Legitimate interests: Processing personal data is necessary for the purposes of the legitimate interests pursued by the Company.

In any case, we will gladly help to clarify the specific legal basis that applies to the processing, and in particular whether the provision of personal data is a statutory (required by law) or a contractual requirement, or a requirement necessary to enter into a contract.

Your Rights under the GDPR

We respect the confidentiality of your personal data and we guarantee that you can exercise your rights.

You have the right under this privacy policy, and by law if you are within the EU or the UK, to:

• Request access to your personal data. The right to access, update or delete the information we have on you. Whenever made possible, you can access, update or request deletion of your personal data directly within your account settings section. If you are unable to perform these actions yourself, please contact us to assist you. This also enables you to receive a copy of the personal data we hold about you.

• Request correction of the personal data that we hold about you. You have the right to have any incomplete or inaccurate information we hold about you corrected.

• Object to processing of your personal data. This right exists where we are relying on a legitimate interest as the legal basis for our processing and there is something about your particular situation, which makes you want to object to our processing of your personal data on this ground. You also have the right to object to us processing your personal data for direct marketing purposes.

• Request erasure of your personal data. You have the right to ask us to delete or remove personal data when there is no good reason for us to continue processing it.

• Request the transfer of your personal data. We will provide to you, or to a third-party you have chosen, your personal data in a structured, commonly used, machine-readable format. Please note that this right only applies to automated information which you initially provided consent for us to use or where we used the information to perform a contract with you.

• Withdraw your consent. You have the right to withdraw your consent on using your personal data. If you withdraw your consent, we may not be able to provide you with access to specific functionalities of our services.

Exercising of Your GDPR Data Protection Rights

You may exercise Your rights of access, rectification, cancellation and opposition by contacting us. Please note that we may ask you to verify your identity before responding to such requests. If you make a request, we will try our best to respond to you as soon as possible.

You have the right to complain to a Data Protection Authority about our collection and use of your personal data. For more information, if you are in the European Economic Area (EEA), please contact Your local data protection authority in the EEA.

Children’s Privacy

Our website may contain content appropriate for children under the age of 13. As a parent, you should know that by using our services, children under the age of 13 may participate in activities that involve the collection or use of personal information. We use reasonable efforts to ensure that before we collect any personal information from a child, the child’s parent receives notice of and consents to our personal information practices.

We may also limit how we collect, use, and store some of the information of users between 13 and 18 years old. In some cases, this means that we will be unable to provide certain functionalities of our services to these users. If we need to rely on consent as a legal basis for processing your information and your country requires consent from a parent, We may require the parent’s consent before we collect and use that information.

We may ask a user to verify their date of birth before collecting any personal information from them. If the user is under the age of 13, the service will either be blocked or redirected to a parental consent process.

Information Collected from Children Under the Age of 13

We may collect and store persistent identifiers, such as cookies or IP addresses from children, without parental consent for the purpose of supporting the internal operation of the our services.

We may collect and store other personal information about children if this information is submitted by a child with prior parental consent or by the parent or guardian of the child.

We may collect and store the following types of personal information about a child when submitted by a child with prior parental consent or by the parent or guardian of the child:

First and/or last name
Date of birth
Gender
Grade level/year group
Email address
Telephone number
Parent’s or guardian’s name
Parent’s or guardian’s email address
For further details on the information we might collect, you can refer to the “Types of Data Collected” section of this privacy policy. We follow our standard privacy policy for the disclosure of personal information collected from and about children.

Parental Access

A parent who has already given the Company permission to collect and use their child’s personal information can, at any time:

Review, correct or delete the child’s personal information
Discontinue further collection or use of the child’s personal information
To make such a request, you can write to us using the contact information provided in this privacy policy.

Links to Other Websites

Our Service may contain links to other websites that are not operated by us. If you click on a third party link, you will be directed to that third party’s site. We strongly advise you to review the privacy policy of every site you visit.

We have no control over and assume no responsibility for the content, privacy policies or practices of any third party sites or services.

Policies Unique to Our Tutoring Services

The following applies to students that have registered for our tutoring services and tutors and employees representing Shalom Education Ltd:

Your profile includes information such as your username and password, purchases or orders made by you, your interests, preferences, and feedback and survey responses.

Booking history including booking requests, subjects booked, set-up test while using our digital resources, any online meeting and recordings of your bookings

Electronic recordings from CCTV, tutoring session, video conference calls, communication history with other users, your interaction with Shalom Education Tutors, records of referral codes shared, redeemed and any promotions that have been applied to your account. All photo’s, videos and conference recordings may be used to protect learner’s, Tutors and employees and may be used as evidence to support any complaint.

Changes to this Privacy Policy

We may update Our Privacy Policy from time to time. We will notify You of any changes by posting the new Privacy Policy on this page.

We may let You know via email and/or a prominent notice on our website prior to the change becoming effective and update the “Last updated” date at the top of this Privacy Policy.

You are advised to review this Privacy Policy periodically for any changes. Changes to this Privacy Policy are effective when they are posted on this page.

Contact Us

If you have any questions about this Privacy Policy, You can contact us:

By email: info@shalom-education.com
By visiting this page on our website: https://www.shalom-education.com/contact
By phone number: 01206657616

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