Question Video: Simplifying an Algebraic Expression Involving Negative and Fractional Exponents Mathematics

Video Transcript

Simplify the expression two 𝑥 to the power of negative one-quarter over three 𝑦 to the power of two-thirds all squared, where 𝑥 and 𝑦 are positive numbers.

We will answer this question using our knowledge of the laws of exponents or indices. We begin by recalling that when we raise a fraction to a power, we can simply raise the numerator and denominator to that power separately. 𝑎 over 𝑏 to the 𝑛th power is equal to 𝑎 to the 𝑛th power over 𝑏 to the 𝑛th power. This means that we can rewrite our expression as two 𝑥 to the power of negative one-quarter squared divided by three 𝑦 to the power of two-thirds squared.

At this stage, we might be tempted to use the power rule of exponents, which states that 𝑎 to the power of 𝑏 raised to the power of 𝑐 is equal to 𝑎 to the power of 𝑏 multiplied by 𝑐. However, since we’re squaring the numerator and denominator, we will simply multiply two 𝑥 to the power of negative one-quarter by itself and also multiply three 𝑦 to the power of two-thirds by itself.

We can now use the rule that states 𝑎 to the power of 𝑏 multiplied by 𝑎 to the power of 𝑐 is equal to 𝑎 to the power of 𝑏 plus 𝑐. On the numerator, we need to add negative one-quarter and negative one-quarter. This is equal to negative two-quarters which simplifies to negative one-half. Two 𝑥 to the power of negative one-quarter multiplied by two 𝑥 to the power of negative one-quarter is therefore equal to four 𝑥 to the power of negative one-half. We can repeat this process on the denominator, giving us a power or exponent of four-thirds. Three 𝑦 to the power of two-thirds squared is equal to nine 𝑦 to the power of four-thirds.

At this stage, we might think we have finished. However, we can also write the term with a negative exponent in a simplified form. 𝑎 to the power of negative 𝑛 is equal to one over 𝑎 to the power of 𝑛. This means that 𝑥 to the power of negative one-half is equal to one over 𝑥 to the power of one-half. And we can therefore rewrite our expression as four over nine 𝑥 to the power of a half 𝑦 to the power of four-thirds. This is the simplified version of two 𝑥 to the power of negative one-quarter over three 𝑦 to the power of two-thirds all squared.