Select The Best Answer For The Question.1. What Is The Value Of $32^4$?A. 648 B. 128 C. $1,048,576$ D. 18

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a clear understanding of the underlying principles. In this article, we will focus on solving exponential equations, specifically the equation $32^4$. We will break down the solution step by step, providing a clear and concise explanation of the process.

Understanding Exponential Equations

Exponential equations involve a base number raised to a power. In the equation $32^4$, the base number is 32, and the power is 4. To solve this equation, we need to understand the concept of exponentiation and how to evaluate expressions with exponents.

Evaluating Exponential Expressions

When evaluating an exponential expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Solving the Equation $32^4$

To solve the equation $32^4$, we need to evaluate the expression using the order of operations.

  1. Exponents: Evaluate the exponential expression $32^4$.
  2. Multiplication and Division: Since there are no multiplication or division operations, we can skip this step.
  3. Addition and Subtraction: Since there are no addition or subtraction operations, we can skip this step.

Calculating the Value

To calculate the value of $32^4$, we need to multiply 32 by itself 4 times:

324=32×32×32×3232^4 = 32 \times 32 \times 32 \times 32

Using a calculator or performing the multiplication manually, we get:

324=1,048,57632^4 = 1,048,576

Conclusion

In conclusion, solving exponential equations requires a clear understanding of the underlying principles and the order of operations. By following the steps outlined in this article, we can evaluate exponential expressions and solve equations like $32^4$. The correct answer to the equation is $1,048,576$.

Answer Options

  • A. 648
  • B. 128
  • C. $1,048,576$
  • D. 18

Correct Answer

The correct answer is C. $1,048,576$.

Discussion

Q: What is an exponential equation?

A: An exponential equation is an equation that involves a base number raised to a power. For example, the equation $32^4$ is an exponential equation, where 32 is the base number and 4 is the power.

Q: How do I evaluate an exponential expression?

A: To evaluate an exponential expression, you need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: What is the difference between an exponential equation and a polynomial equation?

A: An exponential equation involves a base number raised to a power, while a polynomial equation involves a sum of terms with different powers. For example, the equation $x^2 + 3x - 4$ is a polynomial equation, while the equation $2^3$ is an exponential equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to evaluate the expression using the order of operations. If the equation involves a variable, you may need to isolate the variable using algebraic manipulations.

Q: What is the value of $2^{10}$?

A: To evaluate the expression $2^{10}$, you need to multiply 2 by itself 10 times:

210=2×2×2×2×2×2×2×2×2×22^{10} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2

Using a calculator or performing the multiplication manually, you get:

210=1,0242^{10} = 1,024

Q: What is the value of $5^{-2}$?

A: To evaluate the expression $5^{-2}$, you need to divide 1 by 5 squared:

5−2=152=1255^{-2} = \frac{1}{5^2} = \frac{1}{25}

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, you need to combine like terms and apply the rules of exponents. For example, the expression $2^3 \times 2^2$ can be simplified as:

23×22=23+2=252^3 \times 2^2 = 2^{3+2} = 2^5

Q: What is the difference between an exponential function and an exponential equation?

A: An exponential function is a function that involves a base number raised to a power, while an exponential equation is an equation that involves a base number raised to a power. For example, the function $f(x) = 2^x$ is an exponential function, while the equation $2^3 = 8$ is an exponential equation.

Conclusion

In conclusion, exponential equations are a fundamental concept in mathematics, and solving them requires a clear understanding of the underlying principles and the order of operations. By following the steps outlined in this article, you can evaluate exponential expressions and solve equations like $32^4$.