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Chapter 2: Problem 120

Suppose \(x, y,\) and \(z\) are real numbers and \(m\) is a positive integer. Explain why $$ x^{m} y^{m} z^{m}=(x y z)^{m} $$

Short Answer

Expert verified
Using the properties of exponents, we can show that \(x^{m}y^{m}z^{m}=(xyz)^{m}\) as follows: 1. Apply property \((a^m)(b^m) = (ab)^m\) to \(x^{m}y^{m}\), resulting in \((xy)^{m}\). 2. Apply the same property to \(((xy)^{m})(z^{m})\), resulting in \(((xy)z)^{m}\). 3. Simplify the expression inside the parentheses: \(((xy)z)^{m} = (xyz)^{m}\). Thus, it is proven that \(x^{m}y^{m}z^{m} = (xyz)^{m}\).

Step by step solution

01

Understand the properties of exponents

Before we begin, let's understand the properties of exponents relevant to this problem: 1. \((a^m)(b^m) = (ab)^m\) 2. \((a^m)^n = a^{mn}\) These properties hold true for any real numbers a and b, and any positive integers m and n.
02

Apply the first property of exponents

Let's use the first property of exponents, \((a^m)(b^m) = (ab)^m\), to our expression: \(x^{m}y^{m}=(xy)^{m}\)
03

Apply the property of exponents again

Now, let's apply this same property of exponents to our result in step 2 and expression \(z^{m}\) by considering \(a = xy\) and \(b = z\): \(((xy)^{m})(z^{m})=((xy)z)^{m}\)
04

Simplify the expression

Lastly, let's simplify the expression inside the parentheses on the right side of the equation: \(((xy)z)^{m}\) can be rewritten as \((xyz)^{m}\)
05

Combine the steps and write the final result

Putting all the steps together, we have: \(x^{m}y^{m}z^{m} = ((xy)^{m})(z^{m}) = (xyz)^m\) So, it is proven that \(x^{m}y^{m}z^{m} = (xyz)^{m}\).

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