Video Transcript
Is it true that three squared to the fifth power is equal to three to the fifth power squared?
Let’s have a closer look at the statement we’ve been asked to consider. Both sides are exponential expressions of the same rational base, which is three. On the left-hand side, the base is squared, or raised to the second power, and then raised to the fifth power, whereas on the right-hand side, the base is raised to the fifth power and then squared. As both sides involve raising a base to one power and then another, we can recall a law of exponents, which states that when we raise a base to a power and then another power, overall the base is raised to the product of those powers.
The expression on the left-hand side therefore becomes three to the power of two times five. The expression on the right-hand side becomes three to the power of five times two. If we evaluate each product, both exponents become 10. And so we have three to the 10th power on each side. Without evaluating, we know that these two expressions must be equal, because we have the same base raised to the same power. In fact, we didn’t even need to evaluate the powers if we recall that multiplication is commutative. So we know without evaluating that five multiplied by two gives the same result as two multiplied by five. So we can deduce that the expressions in the first stage of our working are of equal value, as they have the same base raised to the same power.
We’ve found that both expressions simplify to three to the 10th power. So our answer to the question “Is it true that these expressions are equal?” is yes.
