Video Transcript
Which of the following expressions
has the same value as 𝑎 over 𝑏 cubed multiplied by 𝑎 over 𝑏 to the ninth power
over 𝑎 over 𝑏 to the eighth power all raised to the fifth power? (A) 𝑎 over 𝑏 to the fourth
power. (B) 𝑎 over 𝑏 to the fifth
power. (C) 𝑎 over 𝑏 to the third power
to the fourth power. (D) 𝑎 over 𝑏 to the third power
to the fifth power. (E) 𝑎 over 𝑏 to the fifth power
to the fourth power.
We’ve been asked which of five
expressions is equivalent to a given expression. Looking more closely at this
expression, we can see that it is an exponential expression involving three powers
of the same base, 𝑎 over 𝑏. Each of the answer options also
involves the same base but is in a simplified form. To answer this question, we’ll need
to simplify this expression using laws of exponents.
In the numerator of the quotient,
we have the product of two powers of this base. We can recall that the product rule
for exponents states that when we multiply powers of the same base together, we add
the powers. So our first step is to combine the
terms in the numerator by applying this rule to give 𝑎 over 𝑏 to the power of
three plus nine, which simplifies to 𝑎 over 𝑏 to the 12th power.
Next, we note that we are now
dividing one power of the base by another. We can therefore recall another law
of exponents, which states that when dividing two powers of the same nonzero base,
we subtract the powers. Applying this law gives 𝑎 over 𝑏
to the power of 12 minus eight, and this is then raised to the fifth power. Simplifying the exponent inside the
parentheses gives 𝑎 over 𝑏 to the fourth power to the fifth power. Finally, we observe that we now
have one power of the base raised to another power.
We can recall one final law of
exponents, which is that if we raise a base to a power and then to another power,
overall the base is raised to the product of those powers. Applying this law gives 𝑎 over 𝑏
to the power of four multiplied by five, which is 𝑎 over 𝑏 to the 20th power.
We’ve now fully simplified this
expression, and we need to determine which of the five options it is equivalent
to. At first glance, it may appear that
it isn’t actually equivalent to any of the five options, as 𝑎 over 𝑏 to the 20th
power isn’t listed. However, if we look at the final
option, we can see that this is very similar to the penultimate stage of our working
out. But the exponents of four and five
have swapped places. If we apply the final law of
exponents we wrote down to option (E), the expression would become 𝑎 over 𝑏 to the
power of five multiplied by four. But as multiplication is
commutative, five multiplied by four is the same as four multiplied by five. So in both cases, the final power
evaluates to 20.
This is actually illustrative of
another rule of exponents, which is that if we raise a base to a power and then to
another power, that’s equivalent to raising the base to the same powers in the
opposite order, because in both cases the result is the base raised to the product
of those powers.
So the correct answer is option
(E). The expression that is equivalent
to the original expression is 𝑎 over 𝑏 to the fifth power to the fourth power.
