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Simplify six to the power of negative three multiplied by three all over two to the power of negative one.
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To begin, itโs sensible to begin by evaluating each of the numbers in this expression which have exponents.
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They are six to the power of negative three and two to the power of negative one.
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But what does a negative power or a negative exponent actually mean?
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Well, a negative exponent tells us to find the reciprocal of that number.
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So ๐ to the power of negative ๐ is equal to one over ๐ to the power of ๐.
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For example, two to the power of negative three is equal to one over two to the power of three.
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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.
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And it works in much a similar way when weโre dealing with fractions.
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Four-thirds to the power of negative two is the same as three-quarters squared.
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And this is because the reciprocal of four-thirds is three-quarters.
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We can then evaluate the numerator.
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Three squared is nine.
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And the denominator, four squared is 16.
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So four-thirds all squared is the same as nine 16ths.
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Letโs use this to evaluate six to the power of negative three.
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The reciprocal of six is one-sixth.
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So six to the power of negative three is one over six cubed.
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We could evaluate six cubed.
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But itโs probably not the most sensible step to take.
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And weโll see why in a moment.
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The reciprocal of two is one-half.
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So two to the power of negative one is one over two to the power of one.
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And since anything to the power of one is just itself, itโs the same as one-half.
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So letโs rewrite our expression piece by piece.
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Itโs one over six cubed multiplied by three all over one-half.
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And next, we recall the fact that this fraction line means divide.
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And to divide by a fraction, we multiply by the reciprocal of that fraction.
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So this is the same as one over six cubed multiplied by three multiplied by two over one.
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Now multiplication is commutative.
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It can be done in any order.
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So we donโt really need these brackets here.
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And we add a denominator of one to the number three.
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We then multiply the numerators.
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One multiplied by three multiplied by two is six.
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And six cubed multiplied by one multiplied by one gives us a denominator of six cubed.
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Now remember, six is the same as six to the power of one.
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And this time, we recall this rule.
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๐ฅ to the power of ๐ divided by ๐ฅ to the power of ๐ is ๐ฅ to the power of ๐ minus ๐.
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When we divide as long as the basis that sees big numbers are the same, we can subtract the exponents.
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So, one minus three is negative two.
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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.
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And weโve already seen that a negative power tells us to find the reciprocal.
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So six to the power of negative two is one over six squared.
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And since six squared is 36, we can say that one over six squared is the same as one 36th.
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And we have fully simplified this expression.
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Six to the power of negative three multiplied by three over two to the power of negative one is the same as one 36th.