 We want to simplify the following expressions completely. The first expression is two
x raised to the fourth power. Let’s first simplify this by expanding, and then we’ll simplify it again using the power property of exponents. Notice we have an exponent of four, which means we have four
factors of the base, which is two x. So this is equal to two x, times two x, times two x, times two x. Once expanded, notice
how we can see we have four factors of two and four factors of x, so we can write this as two to the fourth times x to the fourth, and we can evaluate two to the fourth because the base is two. Two to the fourth is equal
to four factors of two, which equals 16, and
therefore that simplifies to 16 x to the fourth. And two to the fourth is
equal to two times two, times two, times two, which equals 16. Now let’s look at the power
property of exponents. A raised the power of m,
raised the power of n, is equal to a raised
the power of m times n. So when we have powers raised to powers, we multiply the exponents. This product indicates how
many factors of a we have when we raise a to the m to the nth power. So applying the power
property of exponents to this first expression, we need to view two x as two to the first times x to the first. So this is equal to two raised
the power of one times four, times x raised to the
power of one times four, which equals two raised to
the power of four times x raised to the power of four, which again is equal
to 16 x to the fourth. Let’s look at our second example. We have three v to the sixth,
raised to the fifth power. So the base is three v to the sixth, which we need to think
of as three to the first times v to the sixth, and because we have powers of powers we multiply the exponents. This is equal to three raised to the power of one times five, times
v raised to the power of six times five, which is equal to three to the fifth times v to the 30th. And again we can evaluate
three to the fifth because the base is three. Let’s do this on the calculator. Three to the fifth is equal to 243, and therefore this simplifies
to 243 v to the 30th. To evaluate three to the fifth by hand, we’d expand three to the fifth, which is equal to five factors of three, which does equal 243. Next we have negative four
w to the third, cubed, or negative four w to the third
raised to the third power. Notice how here the base is
negative four w to the third. We need to be careful
four as negative four raised to the first power. So because we have powers of powers, we now multiply the exponents. So this is equal to the
base of negative four, raised to the power of one times three, times w raised to the
power of three times three. So this is equal to negative four raised to the power of three, times w raised to the
power of three times three, which equals nine. And again it is important
that we do have parenthesis around the negative four,
because it indicates the base is negative four, not positive four. A base of negative four
raised to the power of three does not mean the same
thing as negative four raised to the third. Here we have three
factors of negative four, which is equal to negative 64. Here we have one negative sign and three factors of positive four. So while the value is still the same, this is negative 64, they
don’t mean the same thing, and in our case because the negative four is inside the parenthesis,
the base is negative four. So this is equal to
negative 64 w to the ninth. And for our last example, notice how we have the base
of negative two c squared d to the 12th raised to the seventh power. We need to view negative
two as negative two raised to the first before we apply the power property of exponents. So this is equal to the
base of negative two raised to the power of one times seven, times c raised to the
power of two times seven, times d raised to the
power of 12 times seven. So we have negative two
raised to the power of seven times c raised to the power
of two times seven is 14, times d raised to the power
of 12 times seven which is 84, so we have d raised to the power of 84. And now we need to evaluate
the base of negative two raised to the seventh power, and let’s do this on the calculator. So open parenthesis negative
two close parenthesis, raised to the power of seven and enter. Which gives us negative 128. Which means the simplified expression is equal to negative 128 c
to the 14th, d to the 84th. I hope you found this helpful.