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

True or false: The simplified form of ((𝑥^( −1/2)𝑦²)/𝑦⁴)⁻² is 𝑥𝑦⁴.

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Video Transcript

True or false: The simplified form of 𝑥 to the power of negative one-half multiplied by 𝑦 squared divided by 𝑦 to the fourth power all raised to the power of negative two is 𝑥 multiplied by 𝑦 to the fourth power.

In order to answer this question, we will simplify the expression given using our knowledge of the laws of exponents or indices.

We begin by noticing there is a term containing 𝑦 on the numerator and denominator of our fraction. One of our rules of exponents states that 𝑎 to the power of 𝑏 divided by 𝑎 to the power of 𝑐 is equal to 𝑎 to the power of 𝑏 minus 𝑐. This means that 𝑦 squared divided by 𝑦 to the fourth power is equal to 𝑦 to the power of two minus four, which is equal to 𝑦 to the power of negative two. This means that we can rewrite our expression as 𝑥 to the power of negative one-half multiplied by 𝑦 to the power of negative two all raised to the power of negative two.

Next, we recall the power rule of exponents. This 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. We can perform this rule on each of our terms. We have 𝑥 to the power of negative one-half raised to the power of negative two. And this is multiplied by 𝑦 to the power of negative two raised to the power of negative two.

We can then multiply the exponents. Negative one-half multiplied by negative two is equal to one. And negative two multiplied by negative two is equal to four as we recall that multiplying two negative numbers gives a positive answer. Our expression therefore simplifies to 𝑥 multiplied by 𝑦 to the fourth power. And we can therefore conclude that the statement is true.

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