### Video Transcript

Which of the following expressions
has the same value as one-third cubed squared? (A) One-third to the sixth power,
(B) one-third to the fifth power, (C) one-third to the ninth power, (D) two over
three cubed, (E) one-third cubed multiplied by one-third squared.

We’ve been asked which of five
expressions has the same value as the given expression. Now we could evaluate this
expression and each of the five options by evaluating the powers and see which
result in the same value. However, it will be easier to
simplify the given expression using the laws of exponents.

We can recall that if we have a
rational base raised to a positive integer exponent and then raise this to another
positive integer exponent, then this is equivalent to raising that base to the
product of those exponents. So applying this law with the base
𝑎 equal to one-third, the first exponent 𝑚 equal to three, and the second exponent
𝑛 equal to two gives one-third to the power of three times two. This of course simplifies to
one-third to the power of six, which is option (A).

We should briefly consider some of
the other options though as they reveal some of the commonly made mistakes when
dealing with exponents.

In option (B), the answer of
one-third to the fifth power is reached by adding the exponents of two and three
instead of multiplying them.

In option (C), the exponent of nine
comes from raising the first exponent to the power of the second; three squared is
nine.

Option (D) doesn’t really reflect a
common misconception surrounding exponents, but perhaps at some point the one in the
numerator has been multiplied by the power of two instead of raised to it.

In option (E), the expression has
been rewritten as a product of terms raised to the exponents of three and two,
instead of raising the base of one-third to the power of three and then to the power
of two.

Using another law of exponents —
which tells us that when we multiply powers of the same base, we add the exponents —
shows that this expression actually simplifies to one-third to the fifth power,
which is the same as option (B).

The correct expression that has the
same value as one-third cubed squared is option (A): one-third to the sixth
power.