Video: Simplifying Numerical Expressions Involving Negative Exponents

Simplify (6^(โˆ’3) ร— 3)/2(^โˆ’1).

03:14

Video Transcript

Simplify six to the power of negative three multiplied by three all over two to the power of negative one.

To begin, itโ€™s sensible to begin by evaluating each of the numbers in this expression which have exponents. They are six to the power of negative three and two to the power of negative one. But what does a negative power or a negative exponent actually mean?

Well, a negative exponent tells us to find the reciprocal of that number. So ๐‘Ž to the power of negative ๐‘ is equal to one over ๐‘Ž to the power of ๐‘. For example, two to the power of negative three is equal to one over two to the power of three. And since two to the power of three is eight, we can say that two to the power of negative three is the same as one-eighth.

And it works in much a similar way when weโ€™re dealing with fractions. Four-thirds to the power of negative two is the same as three-quarters squared. And this is because the reciprocal of four-thirds is three-quarters. We can then evaluate the numerator. Three squared is nine. And the denominator, four squared is 16. So four-thirds all squared is the same as nine 16ths.

Letโ€™s use this to evaluate six to the power of negative three. The reciprocal of six is one-sixth. So six to the power of negative three is one over six cubed. We could evaluate six cubed. But itโ€™s probably not the most sensible step to take. And weโ€™ll see why in a moment. The reciprocal of two is one-half. So two to the power of negative one is one over two to the power of one. And since anything to the power of one is just itself, itโ€™s the same as one-half.

So letโ€™s rewrite our expression piece by piece. Itโ€™s one over six cubed multiplied by three all over one-half. And next, we recall the fact that this fraction line means divide. And to divide by a fraction, we multiply by the reciprocal of that fraction. So this is the same as one over six cubed multiplied by three multiplied by two over one.

Now multiplication is commutative. It can be done in any order. So we donโ€™t really need these brackets here. And we add a denominator of one to the number three. We then multiply the numerators. One multiplied by three multiplied by two is six. And six cubed multiplied by one multiplied by one gives us a denominator of six cubed. Now remember, six is the same as six to the power of one.

And this time, we recall this rule. ๐‘ฅ to the power of ๐‘Ž divided by ๐‘ฅ to the power of ๐‘ is ๐‘ฅ to the power of ๐‘Ž minus ๐‘. When we divide as long as the basis that sees big numbers are the same, we can subtract the exponents. So, one minus three is negative two. And we can see that six to the power of one divided by six to the power of three is six to the power of negative two.

And weโ€™ve already seen that a negative power tells us to find the reciprocal. So six to the power of negative two is one over six squared. And since six squared is 36, we can say that one over six squared is the same as one 36th. And we have fully simplified this expression.

Six to the power of negative three multiplied by three over two to the power of negative one is the same as one 36th.

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