Question Video: Simplifying an Exponential Expression Involving Positive Integer Powers of a Rational Base Mathematics • 8th Grade

Video Transcript

Given that 𝑥 is a rational number, which of the following expressions is equivalent to 𝑥 to the fifth power multiplied by 𝑥 to the fourth power plus 𝑥 to the fourth power squared? (A) 𝑥 to the 20th power plus 𝑥 to the eighth power. (B) 𝑥 to the 20th power plus 𝑥 to the sixth power. (C) 𝑥 to the ninth power plus 𝑥 to the eighth power. (D) 𝑥 to the ninth power plus 𝑥 to the sixth power. (E) 𝑥 to the fourth power multiplied by 𝑥 to the fifth power plus 𝑥 squared.

We are asked which of five expressions is equivalent to the given expression. Let’s start by identifying the different components of this expression. Each part of the expression involves a power of the same base 𝑥, which we’re told is a rational number. There is a product of two powers of 𝑥, and then this is added to a power of 𝑥 raised to another power.

To simplify this expression, we therefore need to recall two of the laws of exponents. The first law is the multiplication law, which tells us that when multiplying powers of the same rational base, we add the exponents. 𝑥 to the 𝑚th power multiplied by 𝑥 to the 𝑛th power is equal to 𝑥 to the 𝑚 plus 𝑛th power. We can use this law to simplify the first part of the expression: 𝑥 to the fifth power multiplied by 𝑥 to the fourth power is 𝑥 to the power of five plus four, which is 𝑥 to the ninth power.

To simplify the second part of the expression, we need to recall the power law of exponents, which tells us that if we raise a rational base 𝑥 to a power and then to another power, overall 𝑥 is raised to the product of those powers. 𝑥 to the 𝑚th power to the 𝑛th power is 𝑥 to the 𝑚𝑛th power. So, the second term can be simplified to 𝑥 to the power of four times two, which is 𝑥 to the eighth power. We’ve now simplified the overall expression to 𝑥 to the ninth power plus 𝑥 to the eighth power. We can’t simplify this expression any further as the two terms have different powers of 𝑥. Looking at the five options we were given, this expression is option (C).

Let’s take a quick look at some of the other options to identify some common mistakes.

In option (A), the second term is correct, but the exponent of the first term is 20 instead of nine. This corresponds to multiplying the exponents of five and four instead of adding them.

In option (B), both terms are incorrect. The same mistake as in option (A) has been made for the first term, and for the second, the two exponents have been added instead of multiplied.

In option (D), the first term is correct, but once again the exponents have been added instead of multiplied when simplifying the second term.

In option (E), it appears as if an attempt has been made to factor the expression by 𝑥 to the fourth power. If this was done correctly, we would obtain 𝑥 to the fourth power multiplied by 𝑥 to the fifth power plus 𝑥 to the fourth power. The error is that the second term in the parentheses has been written as 𝑥 squared instead of 𝑥 to the fourth power, which is a result of confusing whether the exponents should be added or multiplied.

The correct answer is that the expression that is equivalent to 𝑥 to the fifth power multiplied by 𝑥 to the fourth power plus 𝑥 to the fourth power squared is option (C), 𝑥 to the ninth power plus 𝑥 to the eighth power.