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

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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 $4$ multiplied by itself $4$ times, or $4 \times 4 \times 4 \times 4$. When we have multiple exponents in an expression, we can use the rule of multiplication to simplify it. This rule states that when we multiply two numbers with the same base, we add their exponents.

Applying the Rules of Exponents to the Given Expression

In the given expression, $4^4\left(4^{-7}\right)(4^3]$, we have three terms with the same base, which is $4$. To simplify this expression, we can apply the rule of multiplication by adding the exponents. However, we need to be careful with the negative exponent. A negative exponent indicates that we are dealing with a reciprocal. In other words, $4^{-7}$ is equal to $\frac{1}{4^7}$.

Simplifying the Expression

Using the rule of multiplication, we can simplify the expression as follows:

$4^4\left(4^{-7}\right)(4^3] = 4^{4+(-7)+3}$

Evaluating the Exponent

Now, we need to evaluate the exponent by adding the numbers inside the parentheses.

$4^{4+(-7)+3} = 4^{-0}$

Understanding Zero Exponents

A zero exponent indicates that we are dealing with a reciprocal. In other words, $4^0$ is equal to $1$. This is because any number raised to the power of zero is equal to $1$.

Simplifying the Expression Further

Using the fact that $4^0 = 1$, we can simplify the expression as follows:

$4^{-0} = 1$

Conclusion

Therefore, the product of $4^4\left(4^{-7}\right)(4^3]$ is $1$. This means that the correct answer is not among the options provided.

Understanding the Options

Let's take a closer look at the options provided:

A. $\frac{1}{-16}$ B. $\frac{1}{-8}$ C. $\frac{1}{16}$ D. $\frac{1}{8}$

Evaluating the Options

We can evaluate each option by simplifying the expression using the rules of exponents.

Option A

Option A is $\frac{1}{-16}$. To simplify this expression, we can rewrite it as $-1 \times \frac{1}{16}$. However, this is not equal to the product of $4^4\left(4^{-7}\right)(4^3]$.

Option B

Option B is $\frac{1}{-8}$. To simplify this expression, we can rewrite it as $-1 \times \frac{1}{8}$. However, this is not equal to the product of $4^4\left(4^{-7}\right)(4^3]$.

Option C

Option C is $\frac{1}{16}$. To simplify this expression, we can rewrite it as $\frac{1}{2^4}$. However, this is not equal to the product of $4^4\left(4^{-7}\right)(4^3]$.

Option D

Option D is $\frac{1}{8}$. To simplify this expression, we can rewrite it as $\frac{1}{2^3}$. However, this is not equal to the product of $4^4\left(4^{-7}\right)(4^3]$.

Conclusion

Therefore, none of the options provided is equal to the product of $4^4\left(4^{-7}\right)(4^3]$. The correct answer is $1$.

Final Answer

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

Q: What is the rule of multiplication for exponents?

A: The rule of multiplication for exponents states that when we multiply two numbers with the same base, we add their exponents. For example, $a^m \times a^n = a^{m+n}$.

Q: What is a negative exponent?

A: A negative exponent indicates that we are dealing with a reciprocal. In other words, $a^{-n}$ is equal to $\frac{1}{a^n}$.

Q: What is a zero exponent?

A: A zero exponent indicates that we are dealing with a reciprocal. In other words, $a^0$ is equal to $1$. This is because any number raised to the power of zero is equal to $1$.

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

A: To simplify an expression with multiple exponents, we can use the rule of multiplication by adding the exponents. For example, $a^m \times a^n \times a^p = a^{m+n+p}$.

Q: What is the order of operations for exponents?

A: The order of operations for exponents is as follows:

1. Evaluate any expressions inside parentheses.
2. Evaluate any exponents.
3. Multiply and divide from left to right.
4. Add and subtract from left to right.

Q: How do I evaluate an expression with a negative exponent?

A: To evaluate an expression with a negative exponent, we can rewrite it as a fraction. For example, $a^{-n} = \frac{1}{a^n}$.

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

A: A positive exponent indicates that we are dealing with a direct relationship between the base and the exponent. In other words, $a^m$ means $a$ multiplied by itself $m$ times. A negative exponent indicates that we are dealing with a reciprocal. In other words, $a^{-n}$ means $\frac{1}{a^n}$.

Q: How do I simplify an expression with a zero exponent?

A: To simplify an expression with a zero exponent, we can rewrite it as $1$. For example, $a^0 = 1$.

Q: What is the rule for dividing exponents?

A: The rule for dividing exponents states that when we divide two numbers with the same base, we subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

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

A: To evaluate an expression with multiple exponents and fractions, we can use the rule of multiplication and division by adding and subtracting the exponents. For example, $\frac{a^m \times a^n}{a^p} = a^{m+n-p}$.

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

A: An exponent is a small number that is raised to a power. A power is the result of raising a number to an exponent. For example, $a^m$ is an exponent, while $a^m = b$ is a power.

Q: How do I simplify an expression with a fractional exponent?

A: To simplify an expression with a fractional exponent, we can rewrite it as a square root. For example, $a^{\frac{1}{2}} = \sqrt{a}$.

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

A: The rule for raising a power to a power states that when we raise a power to a power, we multiply the exponents. For example, $(a^m)^n = a^{m \times n}$.

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

A: To evaluate an expression with multiple powers and exponents, we can use the rule of multiplication and division by adding and subtracting the exponents. For example, $(a^m)^n \times a^p = a^{m \times n + p}$.

Q: What is the difference between a power and a root?

A: A power is the result of raising a number to an exponent, while a root is the result of taking a number to a fractional exponent. For example, $a^m$ is a power, while $\sqrt{a}$ is a root.

Q: How do I simplify an expression with multiple roots and powers?

A: To simplify an expression with multiple roots and powers, we can use the rule of multiplication and division by adding and subtracting the exponents. For example, $\sqrt{a} \times a^m = a^{\frac{1}{2} + m}$.

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

A: The rule for raising a root to a power states that when we raise a root to a power, we multiply the exponents. For example, $(\sqrt{a})^n = a^{\frac{n}{2}}$.

Q: How do I evaluate an expression with multiple roots and powers?

A: To evaluate an expression with multiple roots and powers, we can use the rule of multiplication and division by adding and subtracting the exponents. For example, $(\sqrt{a})^n \times a^m = a^{\frac{n}{2} + m}$.

Q: What is the difference between a root and a logarithm?

A: A root is the result of taking a number to a fractional exponent, while a logarithm is the result of taking a number to a power. For example, $\sqrt{a}$ is a root, while $\log_a{b}$ is a logarithm.

Q: How do I simplify an expression with multiple roots and logarithms?

A: To simplify an expression with multiple roots and logarithms, we can use the rule of multiplication and division by adding and subtracting the exponents. For example, $\sqrt{a} \times \log_a{b} = a^{\frac{1}{2}} \times \log_a{b}$.

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

A: The rule for raising a logarithm to a power states that when we raise a logarithm to a power, we multiply the exponents. For example, $(\log_a{b})^n = \log_a{b^n}$.

Q: How do I evaluate an expression with multiple logarithms and powers?

A: To evaluate an expression with multiple logarithms and powers, we can use the rule of multiplication and division by adding and subtracting the exponents. For example, $(\log_a{b})^n \times a^m = \log_a{b^n} \times a^m$.

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

A: A logarithm is the result of taking a number to a power, while an exponential function is the result of raising a number to an exponent. For example, $\log_a{b}$ is a logarithm, while $a^b$ is an exponential function.

Q: How do I simplify an expression with multiple logarithms and exponential functions?

A: To simplify an expression with multiple logarithms and exponential functions, we can use the rule of multiplication and division by adding and subtracting the exponents. For example, $\log_a{b} \times a^m = \log_a{b} \times a^m$.

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

A: The rule for raising an exponential function to a power states that when we raise an exponential function to a power, we multiply the exponents. For example, $(a^b)^c = a^{b \times c}$.

Q: How do I evaluate an expression with multiple exponential functions and powers?

A: To evaluate an expression with multiple exponential functions and powers, we can use the rule of multiplication and division by adding and subtracting the exponents. For example, $(a^b)^c \times a^m = a^{b \times c + m}$.

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

A: An exponential function is the result of raising a number to an exponent, while a power function is the result of raising a number to a power. For example, $a^b$ is an exponential function, while $a^m$ is a power function.

Q: How do I simplify an expression with multiple exponential functions and power functions?

A: To simplify an expression with multiple exponential functions and power functions, we can use the rule of multiplication and division by adding and subtracting the exponents. For example, $a^b \times a^m = a^{b + m}$.

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

A: The rule for raising a power function to a power states that when we raise a power function to a power, we multiply the exponents. For example, $(a^m)^n = a^{m \times n}$.

Q: How do I evaluate an expression with multiple power functions and powers?

A: To evaluate an expression with multiple power functions and powers, we can use the rule of multiplication and division by adding and subtracting the exponents. For example, $(a^m)^n \times a^p = a^{m \times n + p}$.

Q: What is the difference between a power function and a root function?

A: A power function is the result of raising a number to a power, while a root function is the result of taking a number to a fractional exponent. For