Which of the following has the same value as one and a quarter to the power of negative two divided by one and a quarter to the power of negative three? Is it (A) four-fifths, (B) five-quarters, (C) five-quarters to the power of three over two, (D) five-quarters to the power of negative five, or (E) the square root of one and a quarter to the power of negative three?
In order to answer this question, we need to recall one of our laws of exponents or indices. 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 minus 𝑏. When dividing two terms with the same base, we can subtract the exponents.
In this question, we need to subtract negative three from negative two. This is the same as adding three to negative two. Negative two plus three is equal to one. So, our expression simplifies to one and a quarter to the power of one. Any number raised to the power of one is itself. So, our answer is one and a quarter. This does not match any of our options, so we need to turn our mixed number into a top-heavy or improper fraction.
We know that in one whole one, we have four-quarters. This means that one and a quarter is the same as five-quarters or five over four. A quick way to calculate the numerator when converting a mixed number is to multiply the whole number by the denominator and then adding the numerator. One multiplied by four plus one is equal to five. Our denominator when converting stays the same.
The correct answer is option (B). One and a quarter to the power of negative two divided by one and a quarter to the power of negative three is five over four or five-quarters.