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Chapter 6: Problem 37

Write with a single exponent. $$ \frac{4^{n}}{2^{m}} $$

Short Answer

Expert verified
Question: Simplify the expression $$\frac{4^n}{2^m}$$. Answer: The simplified expression is $$2^{2n-m}$$.

Step by step solution

01

Rewrite base 4 as a power of 2

First, we need to rewrite the base 4 in terms of base 2, so that both bases are the same and we can apply the rules of exponents. We know that 4 can be written as $$2^2$$, so we have: $$ \frac{4^n}{2^m} = \frac{(2^2)^n}{2^m} $$
02

Apply the power to a power property

Now, we apply the power to a power property, which states that $$(a^n)^m = a^{nm}$$. Using this property, we can simplify the numerator as follows: $$ \frac{(2^2)^n}{2^m} = \frac{2^{2n}}{2^m} $$
03

Apply the division property

Now that we have the same base in both the numerator and denominator, we can apply the division property mentioned earlier, which states that $$a^n / a^m = a^{n-m}$$. Applying this property to our expression, we get: $$ \frac{2^{2n}}{2^m} = 2^{2n-m} $$ So, the simplified expression with a single exponent is: $$ \frac{4^n}{2^m} = 2^{2n-m} $$

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Key Concepts

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

Power to a Power Property
The power to a power property deals with how we manage exponents when one expression with an exponent is raised to another exponent. Imagine stacking exponents on top of each other. It can be expressed in a simple formula: \[ (a^n)^m = a^{n \times m} \] Here, you multiply the exponents.
For instance, with \( (2^2)^n \), you multiply 2 by \( n \), giving \( 2^{2n} \).
  • Stacked exponents multiply each other together.
  • Keep the base the same; only the exponents are multiplied.
  • This helps to simplify complex expressions involving multiple exponents.

This property is useful in simplifying expressions before applying other rules, such as in our example \( \frac{(2^2)^n}{2^m} \). By using the power to a power property, it becomes \( \frac{2^{2n}}{2^m} \), which sets us up nicely for the next step.
Division Property of Exponents
The division property of exponents helps to simplify expressions where two exponential numbers with the same base are divided. The rule for this property is expressed as: \[ \frac{a^n}{a^m} = a^{n-m} \] In simpler terms, subtract the exponent in the denominator from the exponent in the numerator.
For our example, \( \frac{2^{2n}}{2^m} = 2^{2n-m} \). By applying this rule:
  • Ensure the bases are the same.
  • Subtract the bottom exponent from the top one.
  • This simplification keeps calculations straightforward.

This property is useful in simplifying compound fractional exponent expressions, leading to clearer and more efficient calculations.
Simplifying Exponents
Simplifying exponents involves using various properties like the power to power and division property of exponents. The goal is to condense expressions into simpler, frequently more useful forms. This process reduces clutter and complexity by presenting a single term result.
  • Use the power to a power property to handle stacked exponents.
  • Apply the division property for fractions with the same base.
  • Aim for the simplest form possible, reducing the potential for errors.

In the example \( \frac{4^n}{2^m} \), the task is to rewrite it using a single exponent. By converting \( 4 \) to \( 2^2 \) and simplifying, we achieved \( 2^{2n-m} \). This final expression shows the power of applying these properties, making calculations much more manageable.

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Most popular questions from this chapter

The surface area (not including the base) of a right circular cone of radius \(r\) and height \(h>0\) is given by $$ \pi r \sqrt{r^{2}+h^{2}} $$ Explain why the surface area is always greater than \(\pi r^{2}\) (a) In terms of the structure of the expression. (b) In terms of geometry.

Write each expression without parentheses. Assume all variables are positive. $$ 3\left(2^{x} e^{x}\right)^{4} $$

Write with a single exponent. $$ a^{5} b^{5} $$

Without a calculator, decide whether the quantities are positive or negative. $$ (-23)^{42} $$

Find a conjugate of each expression and the product of the expression with the conjugate. $$ \sqrt{13}-10 $$
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