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Write an exponential equation in the form of 𝑦 equals 𝑏 to the power of 𝑥 for the numbers in the table.
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𝑥 equals two.
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𝑦 equals nine sixteenths.
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𝑥 equals four.
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𝑦 equals 81 over 256.
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𝑥 equals five.
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𝑦 equals 243 over 1024.
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We’ve already been told the form that our exponential equation should take.
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It is 𝑦 is equal to 𝑏 to the power of 𝑥.
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So what are we really looking to work out?
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We’re looking to find the value of 𝑏 for the set of numbers listed.
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And to do this, we can choose any pair of values in the table.
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Let’s choose the first two.
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𝑥 is equal to two.
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And 𝑦 is equal to nine sixteenths.
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We’re going to substitute these into the equation given.
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And at this point, it’s important to know that we could do this with any pair of numbers in the table.
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We’ve chosen two and nine sixteenths mainly because they’re such small numbers.
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𝑦 is equal to nine sixteenths.
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And 𝑥 is equal to two.
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So our equation is nine sixteenths is equal to 𝑏 squared.
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To solve this equation for 𝑏, we find the square root of both sides of the equation.
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The square root of 𝑏 is 𝑏.
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So we need to work out the square root of nine sixteenths.
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But remember, finding a square root yields a positive and a negative result.
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To square root nine sixteenths, we simply square root both the numerator and the denominator of the fraction.
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The square root of nine is three.
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And the square root of 16 is four.
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So 𝑏 could be plus or minus three-quarters.
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So how do we decide whether it is a positive or negative?
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Well, this time, we’re going to consider the third entry in the table.
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𝑥 is equal to five when 𝑦 is equal to 243 over 1024.
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Let’s use the positive value of 𝑏 first.
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And we’ll substitute 𝑥 is equal to five into this equation.
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That’s three-quarters to the power of five, which gives us 243 over 1024, which is what we we’re expecting.
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So this works for a positive value of three-quarters.
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Let’s repeat this process with negative three-quarters.
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This time, we have negative three-quarters to the power of five.
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That gives us negative 243 over 1024.
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That’s not the answer we’re looking for.
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So we can say that 𝑏 must be three-quarters.
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And therefore, the equation in exponential form is 𝑦 is equal to three-quarters to the power of 𝑥.
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And in fact, we can perform one final check.
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This time, we’ll substitute the as-yet unused values in the second entry in our table.
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That’s 𝑥 is equal to four.
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And 𝑦 is equal to 81 over 256.
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𝑦 is equal to three-quarters to the power of four, which is indeed 81 over 256.
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And 𝑦 is equal to three-quarters to the power of 𝑥.