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Chapter 5: Problem 23

Simplify each expression using the products-to-powers rule. $$\left(-2 x^{7}\right)^{5}$$

Short Answer

Expert verified
The simplified expression is \(-32*x^{35}\)

Step by step solution

01

Apply the Basic Rule for Exponents

Apply the basic rules for exponents to inside the parenthesis. The rule states \(a^{m^{n}} = a^{mn}\). The product \( -2^{5}\) which is \(-32 \) and the power \( x^{7*5}\) which is \(x^{35}\), so our expression becomes: \(-32*x^{35}\).
02

Simplify the Expression

In this step there is no further simplification needed our exponent is already simplified, so it remains the same: \(-32*x^{35}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Products-to-Powers Rule
The products-to-powers rule is super handy when you need to simplify expressions that have products raised to an exponent. Here's how it works: if you have \(\left(a \cdot b\right)^n\), you can distribute the power of \(n\) to each part separately, making it \(a^n \cdot b^n\).

For our expression \(\left(-2 x^7\right)^5\), this means applying the exponent of \(5\) to both \(-2\) and \(x^7\). This simplifies to two parts:
  • The number part: \((-2)^5\)\
  • The variable part: \(x^{7 \times 5}\)
When these are calculated:
  • \((-2)^5\) becomes \(-32\).
  • \(x^{35}\) remains as it is.
Putting it together gives you \(-32x^{35}\)! It's a straightforward rule that simplifies multiplying powers in no time.
Simplifying Expressions
Simplifying expressions means making them as basic as possible without changing their value. This makes them easier to understand and work with in math. In our example, after applying the products-to-powers rule, we simplified the base parts separately.
  • We found \((-2)^5 = -32\).
  • We calculated \(x^{7 \times 5} = x^{35}\).
After breaking it down, \(-32x^{35}\) is as simple as the expression can get! Remember, the aim is to have no more steps left to handle. All pieces of the expression should be expressed in their most straightforward form.
Basic Rules for Exponents
Understanding basic rules for exponents is key to mastering expressions like these. Here are some important rules:
  • Product of Powers: When you multiply like bases, you add the exponents: \(a^m \cdot a^n = a^{m+n}\).
  • Power of a Power: When raising a power to another power, multiply the exponents: \(\left(a^m\right)^n = a^{mn}\).
  • Power of a Product: You distribute the exponent to each factor in a product: \(\left(a \cdot b\right)^n = a^n \cdot b^n\).
In our problem, we specifically used the "Power of a Product" rule to simplify the expression. Knowing these rules helps you transform complex expressions into manageable forms.

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Most popular questions from this chapter

determine whether each statement 鈥渕akes sense鈥� or 鈥渄oes not make sense鈥� and explain your reasoning. I used two points and a checkpoint to graph \(y=x^{2}-4\)

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After dividing a polynomial by a binomial, explain how to check the answer.

Use a graphing utility to determine whether the divisions have been performed correctly. Graph each side of the given equation in the same viewing rectangle. The graphs should coincide. If they do not, correct the expression on the right side by using polynomial division. Then use your graphing utility to show that the division has been performed correctly. $$\frac{x^{3}+3 x^{2}+5 x+3}{x+1}=x^{2}-2 x+3$$

Will help you prepare for the material covered in the next section. Simplify: \(\frac{\left(2 x^{3}\right)^{4}}{x^{10}}\)
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