Question Video: Evaluating Rational Numbers Raised to a Power | Nagwa Question Video: Evaluating Rational Numbers Raised to a Power | Nagwa

Question Video: Evaluating Rational Numbers Raised to a Power Mathematics • First Year of Preparatory School

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What is 4/11 × 4/11 × 4/11 × 4/11 × 4/11 × 4/11 × 4/11? [A] (4/11)⁻⁷ [B] (4/11)⁷ [C] (4/11)⁹ [D] (7/11)⁴ [E] (28/11)⁷

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Video Transcript

What is four over 11 multiplied by four over 11 multiplied by four over 11 multiplied by four over 11 multiplied by four over 11 multiplied by four over 11 multiplied by four over 11? Is it option (A) four elevenths to the power of negative seven? (B) Four elevenths to the power of seven. (C) Four elevenths to the power of nine. (D) Seven elevenths to the power of four. Or (E) twenty-eight elevenths to the power of seven.

We could evaluate this expression by multiplying all of the numerators and all of the denominators. This would give us the following expression which we could calculate with or without a calculator. However, the five options in this question are given as powers. This means that instead of evaluating the expression, we can simplify by recalling that repeated multiplication can be written in exponential form. In particular, in this question, we are multiplying seven lots of four elevenths. We know that the product of seven lots of four elevenths can be written by raising four elevenths to an exponent of seven. And as such, we can conclude that the correct answer is option (B). The expression in the question is equivalent to four elevenths to the power of seven.

We will now use the information we have seen so far to define a key property of the powers of rational numbers. Since a positive integer power of a rational base is defined by repeated multiplication, we can show that if 𝑛 is a positive integer and 𝑎 over 𝑏 is a rational number, then 𝑎 over 𝑏 to the 𝑛th power is equal to 𝑎 to the 𝑛th power divided by 𝑏 to the 𝑛th power. In other words, we can raise the numerator and denominator to the power separately.

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