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Chapter 5: Problem 73

Simplify: \(\frac{\left(2 x^{3}\right)^{4}}{x^{10}}\).

Short Answer

Expert verified
The simplified expression of \(\frac{(2 x^{3})^{4}}{x^{10}}\) is \(16x^{2}\).

Step by step solution

01

Apply the Power of a Power Rule

The Power of a Power Rule states that \((a^{m})^{n} = a^{m*n}\). So in our problem, \( (2x^{3})^{4} = 2^{4} * (x^{3*4}) = 16x^{12}\).
02

Divide the term

Next, we'll handle the division by \(x^{10}\). According to Quotient of Powers Rule, we subtract the exponent in the denominator from the exponent in the numerator. Thus, \( \frac{16 x^{12}}{x^{10}} = 16 x^{(12-10)} = 16 x^{2}\).
03

Final Simplification

Our expression has been fully simplified at this step. The exercise asks us to simplify the given expression which we have done by applying the laws of exponents.

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Key Concepts

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

Understanding the Power Rule
When working with exponents, the **Power of a Power Rule** is incredibly useful. This rule helps us simplify expressions where a power is raised to another power.

In mathematical terms, the rule states: \((a^m)^n = a^{m \times n}\). Basically, you multiply the exponents.

For example, in the exercise given, \((2x^3)^4\), we need to apply this rule:
  • Raise each base inside the parentheses to the power of the outside exponent.
  • For \(2\), it becomes \(2^4 = 16\).
  • For \(x^3\), it turns into \(x^{3 \times 4} = x^{12}\).
Which results in \(16x^{12}\). This step condenses many elements into a simpler form, making calculations easier!
Applying the Quotient of Powers Rule
After applying the power rule, the next step often involves dividing terms with the same base. The **Quotient of Powers Rule** is designed for this. It tells us how to divide exponents with the same base, indicating that you subtract the exponents.

Mathematically, it looks like this: \(\frac{a^m}{a^n} = a^{m-n}\). This rule is straightforward but powerful!

In our problem, after simplifying with the Power Rule, we have \(\frac{16x^{12}}{x^{10}}\). Here's how we apply the rule:
  • Keep the base \(x\).
  • Subtract the exponent in the denominator (10) from the exponent in the numerator (12): \(12 - 10 = 2\).
This results in \(16x^2\). This step helps simplify complex expressions into something that's much easier to work with.
Simplification Process
The final goal of many mathematical exercises is simplification. This makes expressions easier to understand and solve.

During simplification, we apply various rules鈥攍ike the Power Rule and Quotient of Powers鈥攖o break down expressions.

In our exercise, after all simplifications, we find ourselves with the expression \(16x^2\). The process involved:
  • Using the Power Rule to multiply exponents in \((2x^3)^4\).
  • Applying the Quotient of Powers to divide \(x^{12}\) by \(x^{10}\).
By following these steps, we've streamlined the initial complex expression into a neat and simple form, making it handy for further calculations or interpretations. Remember, effective simplification not only solves problems but also enhances comprehension!

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Most popular questions from this chapter

Use a vertical format to find each product. $$\begin{aligned}&4 z^{3}-2 z^{2}+5 z-4\\\&3 z-2\end{aligned}$$

Perform the indicated computations. Express answers in scientific notation. $$\frac{\left(1.2 \times 10^{6}\right)\left(8.7 \times 10^{-2}\right)}{\left(2.9 \times 10^{6}\right)\left(3 \times 10^{-3}\right)}$$

Explain how to simplify an expression that involves a quotient raised to a power. Provide an example with your explanation.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$5^{2} \cdot 5^{-2}>2^{5} \cdot 2^{-5}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I used the product rule for exponents to multiply \(x^{7}\) and \(y^{9}\)
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