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Express the square root of nine π₯ squared over 100 π¦ to the power of six π§ to the power of four in its simplest form.
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In order to solve this problem, weβre gonna have a look at a couple of surd rules.
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Our first rule that weβre going to use is that root π over π is equal to root π over root π.
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Iβm gonna use this to express the square root in our question in a different form.
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You can now see that weβre actually express our square root as root nine π₯ squared over root 100π¦ to the power of six π§ to the power of four.
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Weβre gonna have a look at it in two parts.
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Iβm gonna start with root nine π₯ squared.
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To simplify root nine π₯ squared, we can actually use this surd rule, which is that root π multiplied by root π is equal to root ππ.
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So therefore, we can express root nine π₯ squared as root nine multiplied by root π₯ squared.
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Now, that suggests that you need to use this step every time, but I just wanted to break it down to steps to see how you get to the final answer.
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This gives us the answer of three π₯ cause we get the three because the square root of nine is equal to three.
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And we get the π₯ because the square root of π₯ squared is equal to π₯.
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And strictly, the square root means only the positive root.
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So to ensure that this is the case, you put these vertical lines either side of our answer.
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And what these lines mean is the modulus or absolute value, which will give us the positive value only.
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We can now have a look at the denominator which is root 100π¦ to the power of six π§ to the power of four.
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Again, to enable us to show how we simplify this, Iβve actually split it down to root 100 multiplied by root π¦ to the power of six π§ to the power of four.
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Again, I use root π multiplied by root π equals root ππ to enable me to do this.
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In order to simplify it further, weβre going to introduce a couple of exponent rules.
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The first of which being the πth root of π is equal to π to the power of one over π, which we can use to give us 10 multiplied by π¦ to the power of six π§ to the power of four on parentheses to the power of a half.
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We can then use the second exponent rule, which is π to the power of π in parentheses to the power of π is equal to π to the power of ππ, which means that we actually multiply the powers together.
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So this is gonna allow us to simplify even further.
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So weβre gonna get 10π¦.
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And then, it is the π¦ to the power of six multiplied by a half, which gives us π¦ cubed or π¦ to the power of three.
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And then itβs gonna be π§ to the power of four multiplied by a half, which give us π§ squared.
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And again, we include our vertical lines either side as we want the absolute value or modulus.
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So now, we can move back over to the left-hand side, where weβre gonna get a final answer, where we can say that if we express the square root of nine π₯ squared over 100 π¦ to the power of six π§ to the power of four in its simplest form, weβre gonna get the modulus of three π₯ over 10 π¦ cubed π§ squared.