Question Video: Simplifying an Algebraic Expression Involving Negative Exponents Mathematics

Simplify the expression (2𝑥⁻²𝑦⁵)⁻².

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Video Transcript

Simplify the expression two 𝑥 to the power of negative two 𝑦 to the power of five all raised to the power of negative two.

In order to answer this question, we need to recall some of the laws of exponents or indices. Firstly, we recall that 𝑎𝑏 raised to the power of 𝑐 is equal to 𝑎 to the power of 𝑐 multiplied by 𝑏 to the power of 𝑐. This means that we can split the expression inside the parentheses or brackets. The expression can be rewritten as two to the power of negative two multiplied by 𝑥 to the power of negative two raised to the power of negative two multiplied by 𝑦 to the fifth power 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. In the second term in our expression, we can multiply negative two by negative two. This is equal to four. So this part simplifies to 𝑥 to the fourth power or 𝑥 to the power of four. We can do the same with the 𝑦-part of our expression. Five multiplied by negative two is negative 10. So we have 𝑦 to the power of negative 10. The expression simplifies to two to the power of negative two multiplied by 𝑥 to the power of four multiplied by 𝑦 to the power of negative 10.

Next, we recall what happens when we have a negative exponent. 𝑎 to the power of negative 𝑛 is equal to one over 𝑎 to the power of 𝑛. We simply find the reciprocal. Two to the power of negative two is equal to one over two squared. The third part of our expression can be written as one over 𝑦 to the 10th power. Two squared is equal to four. And then multiplying the three parts of our expression gives us 𝑥 to the fourth power over four 𝑦 to the 10th power. This is the simplified form of the expression two 𝑥 to the power of negative two 𝑦 to the fifth power all raised to the power of negative two.

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