Question Video: Evaluating Numerical Expressions Using Laws of Exponents | Nagwa Question Video: Evaluating Numerical Expressions Using Laws of Exponents | Nagwa

Question Video: Evaluating Numerical Expressions Using Laws of Exponents Mathematics • First Year of Preparatory School

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Calculate ((−3(1/5))⁷ × (−1(1/2))⁶)/((−16/5)⁶ × (−3/2)⁴), giving your answer in its simplest form.

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Video Transcript

Calculate negative three and one-fifth to the seventh power multiplied by negative one and a half to the sixth power all over negative 16 over five to the sixth power multiplied by negative three over two to the fourth power, giving your answer in its simplest form.

We could evaluate this expression by evaluating each power in turn and then multiplying the fractions. However, it will be simpler to use some laws of exponents to simplify the expression before evaluating any powers.

Let’s begin by rewriting each mixed number as a fraction. Negative three and one-fifth is equal to negative 16 over five. And negative one and a half is equal to negative three over two. We now see that we actually have common bases in the numerator and denominator of this quotient. As we are dividing powers of the same rational bases, we can recall the division law of exponents. This states that when we divide powers of the same rational base, we subtract the powers. 𝑎 to the 𝑚th power divided by 𝑎 to the 𝑛th power is 𝑎 to the power of 𝑚 minus 𝑛.

Applying this law to each pair of common bases gives negative 16 over five to the power of seven minus six multiplied by negative three over two to the power of six minus four. Simplifying the exponents gives negative 16 over five to the first power multiplied by negative three over two squared.

Now, as the base in the first term is raised to the first power, the whole term is just equal to the base itself. For the second term, we can recall that when we raise a fraction with a nonzero denominator to a power, this is equivalent to raising the numerator and denominator separately to that power. So, the second term in the product can be written as negative three squared over two squared. Evaluating the squares gives nine over four.

Before we multiply the two fractions, we can simplify by cross canceling a factor of four to give negative four over five multiplied by nine over one. Finally, multiplying using the usual rules of fractions gives negative 36 over five. So, by first using the laws of exponents to simplify the given expression, we’ve found that its value is negative 36 over five.

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