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Evaluate the square root of two to the power of negative four multiplied by 512 to the power of negative a third over eight to the power of negative three.
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Well, if we take a look at the expression we’ve got in this problem, we can see that each of the terms has a different base number.
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That’s the number that is being raised to the power or exponent.
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However, what we want is we want these all to be the same so that we can use our exponent rules.
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Well, we can rewrite it with each term having the same base because 512 is equal to two to the power of nine and eight is equal to two cubed or two to the power of three.
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So if we use this, what we can do is rewrite our expression as the square root of two to the power of negative four multiplied by two to the power of nine to the power of negative a third all divided by two cubed to the power of negative three.
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So now that we have the same base across each of our terms, what we can do is apply our exponent rules.
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So let’s remind ourself of three of those.
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So firstly, we have 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏.
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This is equal to 𝑥 to the power of 𝑎 plus 𝑏.
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Then we have 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏, which is equal to 𝑥 to the power of 𝑎 minus 𝑏.
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And then, finally, we have 𝑥 to the power of 𝑎 all to the power of 𝑏, which is equal to 𝑥 to the power of 𝑎𝑏.
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So we multiply the exponents.
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And then we also have something that is not necessarily a rule in itself but something that we know.
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And that is that the square root of 𝑥 can also be written in exponent form as 𝑥 to the power of a half.
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So first of all, what we’re gonna do is we’re gonna use the third rule that we looked at and also the fact that we know that the square root of 𝑥 is equal to 𝑥 to the power of a half to rewrite our expression.
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So what we’ve got is two to the power of negative four multiplied by two to the power of negative three.
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And that’s because if we have nine multiplied by negative a third, then this is equal to negative nine over three, which is the same as negative three.
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And then on the denominator, we’ve got two to the power of negative nine.
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And that’s cause we had two to the power of three.
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And then we have to raise that to the power of negative three.
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So we multiply three and negative three, which gives us negative nine.
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Then this is all to the power of a half cause we used the identity we know, which is the fact that root 𝑥 is the same as 𝑥 to the power of a half.
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And now if we apply the first rule that we looked at, which was that 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏 equals 𝑥 to the power of 𝑎 plus 𝑏, so adding the exponents, we’re gonna get two to the power of negative seven.
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And that’s because if we add negative four and negative three, we get negative seven, all over two to the power of negative nine.
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And this is to the power of a half.
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So now for the next stage, there’s a couple of ways we can go about this.
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So I’m gonna have a look at both methods.
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The first method is gonna use our second exponent rule first.
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Well, with the first method, what we do is we subtract the exponents because we’re dividing.
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So it’s two to the power of negative seven minus negative nine all to the power of a half, which is gonna give us two squared to the power of a half.
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Well, then we can apply the third rule again because what we do is multiply the exponents.
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We have two multiplied by a half, which is just one.
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So this is gonna give us two to the power of one.
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Well, two to the power of one is just two.
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So therefore, we’ve solved the problem and evaluated our expression.
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However, we did say that we’d look at another method, which we’ll show you now.
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Well, if we have a square root or exponent of a fraction, then the way that we can deal with this is by dealing with the numerator and denominator separately.
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So we got two to the power of negative seven to the power of a half and two to the power of negative nine to the power of a half.
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So then, if we apply our third rule once again, what we’ll get is two to the power of negative seven over two divided by two to the power of negative nine over two.
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So then applying the second rule, we’re gonna get two to the power of negative seven over two minus negative nine over two.
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Well, if you add nine over two to negative seven over two, we’ll just get two to the power of two over two, which is just two, the same answer that we got previously.