Which Expression Is Equivalent To $\left(8^5\right)^4$?A. $8^1$ B. $8^9$ C. $8^{17}$ D. $8^{20}$

Introduction

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. In this article, we will explore the concept of equivalent expressions involving exponents and apply it to solve a specific problem. We will examine the expression $\left(8^5\right)^4$ and determine which of the given options is equivalent to it.

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication of a number. For example, $2^3$ can be read as "2 to the power of 3" and is equivalent to $2 \times 2 \times 2$. The exponent, in this case, is 3, and the base is 2.

Properties of Exponents

There are several properties of exponents that we need to understand to solve this problem. These properties include:

• Product of Powers Property: When multiplying two powers with the same base, we add the exponents. For example, $a^m \times a^n = a^{m+n}$.
• Power of a Power Property: When raising a power to another power, we multiply the exponents. For example, $(a^m)^n = a^{m \times n}$.
• Zero Exponent Property: Any non-zero number raised to the power of 0 is equal to 1. For example, $a^0 = 1$.

Applying the Properties of Exponents

Now that we have a good understanding of the properties of exponents, let's apply them to the given expression $\left(8^5\right)^4$. Using the Power of a Power Property, we can rewrite this expression as $8^{5 \times 4}$.

Simplifying the Expression

Using the Product of Powers Property, we can simplify the expression $8^{5 \times 4}$ to $8^{20}$.

Conclusion

In conclusion, the expression $\left(8^5\right)^4$ is equivalent to $8^{20}$. This is because we can apply the Power of a Power Property to rewrite the expression as $8^{5 \times 4}$, and then simplify it using the Product of Powers Property.

Answer

The correct answer is D. $8^{20}$.

Additional Examples

Here are a few additional examples to help reinforce the concept:

• $\left(3^4\right)^3 = 3^{4 \times 3} = 3^{12}$
• $\left(2^6\right)^2 = 2^{6 \times 2} = 2^{12}$
• $\left(5^3\right)^5 = 5^{3 \times 5} = 5^{15}$

Practice Problems

Here are a few practice problems to help you apply the concept:

• $\left(4^2\right)^3 = ?$
• $\left(6^4\right)^2 = ?$
• $\left(9^3\right)^4 = ?$

Solutions

• $\left(4^2\right)^3 = 4^{2 \times 3} = 4^6$
• $\left(6^4\right)^2 = 6^{4 \times 2} = 6^8$
• $\left(9^3\right)^4 = 9^{3 \times 4} = 9^{12}$
  Frequently Asked Questions (FAQs) =====================================

Q: What is the difference between a base and an exponent?

A: The base is the number being raised to a power, and the exponent is the power to which the base is being raised. For example, in the expression $2^3$, the base is 2 and the exponent is 3.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can use the properties of exponents, such as the Product of Powers Property and the Power of a Power Property. For example, to simplify the expression $\left(8^5\right)^4$, you can use the Power of a Power Property to rewrite it as $8^{5 \times 4}$, and then simplify it using the Product of Powers Property.

Q: What is the zero exponent property?

A: The zero exponent property states that any non-zero number raised to the power of 0 is equal to 1. For example, $a^0 = 1$.

Q: How do I evaluate an expression with multiple exponents?

A: To evaluate an expression with multiple exponents, you can use the properties of exponents, such as the Product of Powers Property and the Power of a Power Property. For example, to evaluate the expression $\left(2^3\right)^4 \times \left(3^2\right)^3$, you can use the Power of a Power Property to rewrite it as $2^{3 \times 4} \times 3^{2 \times 3}$, and then simplify it using the Product of Powers Property.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base is being raised to a power, while a negative exponent indicates that the base is being raised to a power and then taking the reciprocal. For example, $a^3$ is different from $a^{-3}$.

Q: How do I simplify an expression with negative exponents?

A: To simplify an expression with negative exponents, you can use the property that $a^{-n} = \frac{1}{a^n}$. For example, to simplify the expression $2^{-3}$, you can rewrite it as $\frac{1}{2^3}$.

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

A: An exponential expression is an expression that involves a base raised to a power, while a polynomial expression is an expression that involves variables and coefficients raised to various powers. For example, $2^3$ is an exponential expression, while $x^2 + 3x + 2$ is a polynomial expression.

Q: How do I compare exponential expressions?

A: To compare exponential expressions, you can compare the bases and the exponents. For example, to compare the expressions $2^3$ and $3^2$, you can see that the bases are different, so the expressions are not equal.

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

A: An exponential function is a function that involves a base raised to a power, while a polynomial function is a function that involves variables and coefficients raised to various powers. For example, $f(x) = 2^x$ is an exponential function, while $f(x) = x^2 + 3x + 2$ is a polynomial function.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or a graphing software. You can also use the properties of exponents to determine the behavior of the function. For example, to graph the function $f(x) = 2^x$, you can see that the function is increasing as x increases.

Q: What is the difference between an exponential growth and an exponential decay?

A: An exponential growth is a situation where a quantity increases exponentially over time, while an exponential decay is a situation where a quantity decreases exponentially over time. For example, the population of a city may grow exponentially over time, while the amount of a radioactive substance may decay exponentially over time.