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Question Video: Applying Laws of Exponents to Simplify an Expression Involving Rational Bases Mathematics • 8th Grade

If π‘₯ = 1/5, 𝑦 = 1/2, and 𝑧 = 2/5, find the value of π‘₯⁴ Γ· (𝑦𝑧)Β².

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Video Transcript

If π‘₯ equals one-fifth, 𝑦 equals one-half, and 𝑧 equals two-fifths, find the value of π‘₯ to the fourth power divided by 𝑦𝑧 squared.

We’ve been asked to evaluate this expression, which involves powers, products, and division of three fractions. Let’s begin by substituting the values of π‘₯, 𝑦, and 𝑧 into the expression. This gives one-fifth to the fourth power divided by one-half multiplied by two-fifths all squared. We can then evaluate the product. One-half multiplied by two-fifths is equal to one-fifth. So the entire expression simplifies to one-fifth to the fourth power divided by one-fifth squared.

We now observe that we are dividing two powers of the same base. In this case, the base is the fraction one-fifth. We can recall the division law for exponents, which tells us that when we are dividing two powers of a nonzero base, we subtract the exponents. So π‘Ž to the π‘šth power divided by π‘Ž to the 𝑛th power is π‘Ž to the π‘š minus 𝑛th power. Applying this law with π‘Ž equal to one-fifth, π‘š equal to four, and 𝑛 equal to two gives one-fifth to the power of four minus two, which simplifies to one-fifth squared.

To simplify further, we need to evaluate one-fifth squared, which we can do in two ways. One method would simply be to calculate one-fifth multiplied by one-fifth. Recalling that when we multiply fractions, we multiply the numerators together and multiply the denominators together, gives one over 25. Alternatively, we could recall another law of exponents, which is that when we raise a fraction to a power, this is equivalent to raising the numerator and denominator separately to that power. So one-fifth squared is the same as one squared over five squared. And evaluating the squares gives one over 25 again.

Both methods of course give the same result, which is that if π‘₯ equals one-fifth, 𝑦 equals one-half, and 𝑧 equals two-fifths, then the value of π‘₯ to the fourth power divided by 𝑦𝑧 squared is one over 25.

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