Question Video: Evaluating Decimals Raised to Negative Exponents

What is the value of (0.2)^(−3)?

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Video Transcript

What is the value of 0.2 all to the power of negative three?

Well, if we think about 0.2, what the 0.2 is is the same as two-tenths. And that’s because if we look at the place values, our zero is the units and our two is the tenths. So, we’re gonna say that 0.2 is equal to two-tenths. But can we make this any simpler? Well, yes, we can simplify the fraction by dividing the numerator and denominator by two. And when we do this, what we get is one over five or one-fifth because two divided by two is one and 10 divided by two is five. So, we know that 0.2 is equal to a fifth. And in fact, this is a decimal conversion that we should know.

So, what we’re gonna get is one-fifth to the power of negative three. Well, if we recall that we got 𝑥 to the power of negative 𝑛 equals one over 𝑥 to the power of 𝑛, it’s slightly different here because we’ve got a fraction, one over five. But it’s exactly the same theory because what we’re going to do is we’re going to find the reciprocal of one over five. And then we’re gonna put it to the power of three.

So, we can think of it as five cubed over one cubed. And that’s because the reciprocal of one over five is five over one. But usually, we wouldn’t bother writing the one cubed on the denominator. Well, five cubed means five multiplied by five multiplied by five. Well, five multiplied by five is 25, and 25 multiplied by five is 125. So therefore, we can say that the value of 0.2 to the power of negative three is 125.

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