What Is The Product Of $4 4\left(4 {-7}\right)(4)$?A. $\frac{1}{-16}$ B. $ 1 − 8 \frac{1}{-8} − 8 1 [/tex] C. $\frac{1}{16}$ D. $\frac{1}{8}$

Mar 8, 2025 by ADMIN 155 views

$What is the Product of $4^4\left(4^{-7}\right)(4)$?$

Understanding Exponents and Their Rules

When dealing with exponents, it's essential to understand the rules that govern their behavior. Exponents are a shorthand way of representing repeated multiplication. For example, $4^4$ means $4 \times 4 \times 4 \times 4$, which equals $256$. The exponent $4$ tells us how many times to multiply the base $4$.

Applying Exponent Rules to the Given Expression

The given expression is $4^4\left(4{-7}\right)(4)$. To simplify this expression, we need to apply the rules of exponents. When multiplying numbers with the same base, we add their exponents. In this case, we have $4^4$ and $4^{-7}$, both of which have the base $4$. We can rewrite the expression as $4^{4+(-7)}(4)$.

Simplifying the Exponent Expression

Now, let's simplify the exponent expression $4^{4+(-7)}$. When we add $4$ and $-7$, we get $-3$. Therefore, the expression becomes $4^{-3}(4)$.

Applying the Rule for Negative Exponents

When we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. In this case, $4^{-3}$ is equivalent to $\frac{1}{4^3}$. Therefore, the expression becomes $\frac{1}{4^3}(4)$.

Simplifying the Expression Further

Now, let's simplify the expression $\frac{1}{4^3}(4)$. We can rewrite $4^3$ as $64$. Therefore, the expression becomes $\frac{1}{64}(4)$.

Applying the Rule for Multiplying Numbers with the Same Base

When multiplying numbers with the same base, we add their exponents. In this case, we have $\frac{1}{64}$ and $4$. We can rewrite the expression as $\frac{4}{64}$.

Simplifying the Fraction

Now, let's simplify the fraction $\frac{4}{64}$. We can divide both the numerator and the denominator by their greatest common divisor, which is $4$. Therefore, the fraction becomes $\frac{1}{16}$.

Conclusion

In conclusion, the product of $4^4\left(4{-7}\right)(4)$ is $\frac{1}{16}$. This is the correct answer among the given options.

Final Answer

The final answer is: $\boxed{\frac{1}{16}}$

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

A: An exponent is a small number that is written to the upper right of a number or a variable, indicating how many times to multiply the base number or variable. A power, on the other hand, is the result of raising a number or variable to a certain exponent.

Q: What is the rule for multiplying numbers with the same base?

A: When multiplying numbers with the same base, we add their exponents. For example, $4^4 \times 4^3 = 4^{4+3} = 4^7$.

Q: What is the rule for dividing numbers with the same base?

A: When dividing numbers with the same base, we subtract their exponents. For example, $4^4 \div 4^3 = 4^{4-3} = 4^1$.

Q: What is the rule for raising a power to a power?

A: When raising a power to a power, we multiply the exponents. For example, $(4⁴⁾3 = 4^{4 \times 3} = 4^{12}$.

Q: What is the rule for negative exponents?

A: When we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. For example, $4^{-3} = \frac{1}{4^3}$.

Q: What is the rule for zero exponents?

A: When the exponent is zero, the result is always 1. For example, $4^0 = 1$.

Q: What is the rule for fractional exponents?

A: When we have a fractional exponent, we can rewrite it as a root. For example, $4^{1/2} = \sqrt{4} = 2$.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, we need to apply the rules of exponents. We can start by combining like terms, then simplify the expression by applying the rules for multiplying, dividing, raising to a power, and taking reciprocals.

Q: What are some common mistakes to avoid when working with exponents?

A: Some common mistakes to avoid when working with exponents include:

Forgetting to apply the rules of exponents
Not simplifying the expression enough
Making errors when multiplying or dividing numbers with the same base
Not recognizing when to use negative exponents or zero exponents

Q: How do I practice working with exponents?

A: To practice working with exponents, you can try the following:

Start with simple expressions and gradually work your way up to more complex ones
Use online resources or worksheets to practice applying the rules of exponents
Try solving problems on your own, then check your answers with a calculator or a teacher
Join a study group or work with a partner to practice working with exponents together

Q: What are some real-world applications of exponents?

A: Exponents have many real-world applications, including:

Science: Exponents are used to describe the growth or decay of populations, the spread of diseases, and the behavior of chemical reactions.
Finance: Exponents are used to calculate interest rates, investment returns, and the value of stocks and bonds.
Engineering: Exponents are used to describe the behavior of electrical circuits, the strength of materials, and the performance of engines.
Computer Science: Exponents are used to describe the growth of algorithms, the behavior of data structures, and the performance of computer systems.

Q: How do I know when to use exponents in a real-world problem?

A: To know when to use exponents in a real-world problem, you need to look for situations where you are dealing with repeated multiplication or division. For example, if you are calculating the growth of a population over time, you may need to use exponents to describe the rate of growth. If you are calculating the interest on a loan, you may need to use exponents to describe the rate of interest.