Question Video: Evaluating an Exponential Expression Involving Positive and Negative Integer Exponents | Nagwa Question Video: Evaluating an Exponential Expression Involving Positive and Negative Integer Exponents | Nagwa

Question Video: Evaluating an Exponential Expression Involving Positive and Negative Integer Exponents Mathematics • Second Year of Preparatory School

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If π‘₯ = 1 and 𝑦 = √2, find (π‘₯⁻² 𝑦⁴)⁻³ in its simplest form.

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Video Transcript

If π‘₯ is equal to one and 𝑦 is equal to root two, find π‘₯ to the power of negative two 𝑦 to the fourth power all to the power of negative three in its simplest form.

Let’s begin by substituting the values of π‘₯ and 𝑦 into the exponential expression. Doing so gives one to the power of negative two multiplied by root two to the fourth power all raised to the power of negative three.

We now need to simplify this expression. First, we can recall that the value one raised to any power is just one. So in particular, one to the power of negative two is one. And the first term in the product simplifies to one. Next, we can recall that raising a base to a positive integer power means we multiply that number of bases together. So root two to the fourth power means four lots of root two multiplied together.

We can then recall that for nonnegative values of π‘Ž, the square root of π‘Ž squared is equal to π‘Ž. So, grouping the four root twos into two pairs gives root two squared multiplied by root two squared. Applying the law we’ve just discussed, each of these factors simplifies to two. So the overall product simplifies to two multiplied by two, which is four.

We can now substitute the values of one to the power of negative two and root two to the fourth power back into the expression. And it becomes one multiplied by four all to the power of negative three. Of course, one multiplied by four is just four. So the expression simplifies to four to the power of negative three. Next, we recall that negative exponents define reciprocals. For a nonzero base π‘Ž, π‘Ž to the power of negative π‘š is equal to one over π‘Ž to the π‘šth power. Four to the power of negative three can therefore be written as one over four to the third power.

Finally, we evaluate four to the third power, or four cubed, by recalling that this is equal to four multiplied by four multiplied by four. We should recall from memory that four cubed is equal to 64. Or we can work it out by first multiplying four by four to give 16 and then multiplying this by four again to give 64. Substituting this value back into the denominator gives our answer.

By applying laws of exponents, we’ve found that the value of the given expression, in its simplest form, is one over 64.

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