Simplify The Expression And Express Your Answer: ( 8 5 ) 7 \left(8^5\right)^7 ( 8 5 ) 7 □ \square □

Understanding Exponents and Power Rules

In mathematics, exponents and power rules are essential concepts that help us simplify complex expressions. When dealing with exponents, it's crucial to understand the rules that govern their behavior. In this article, we will focus on simplifying the expression $\left(8^5\right)^7$ and express our answer.

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication. For example, $2^3$ can be read as "2 to the power of 3" or "2 cubed." It means that 2 is multiplied by itself 3 times: $2^3 = 2 \times 2 \times 2 = 8.$

Power Rule for Exponents

The power rule for exponents states that when we raise a power to another power, we multiply the exponents. In other words, $(a^m)^n = a^{m \times n}.$ This rule helps us simplify expressions with multiple exponents.

Simplifying the Expression

Now that we understand the power rule for exponents, let's apply it to the given expression $\left(8^5\right)^7.$ Using the power rule, we can simplify the expression as follows:

$\left(8^5\right)^7 = 8^{5 \times 7} = 8^{35}$

Calculating the Value

To calculate the value of $8^{35},$ we can use a calculator or estimate the value using the properties of exponents. Since $8^1 = 8$ and $8^2 = 64,$ we can see that $8^3 = 8 \times 8 \times 8 = 512.$

Using this pattern, we can estimate the value of $8^{35}$ as follows:

$8^{35} = 8^{30} \times 8^5 = (8^3)^{10} \times 8^5 = 512^{10} \times 8^5$

Since $512^{10}$ is a very large number, we can approximate it as follows:

$512^{10} \approx 10^{30}$

Now, we can multiply this value by $8^5$ to get:

$8^{35} \approx 10^{30} \times 8^5 = 10^{30} \times 32768$

Using a calculator, we can calculate the value of $8^{35}$ as follows:

$8^{35} \approx 9.33262154439 \times 10^{35}$

Conclusion

In this article, we simplified the expression $\left(8^5\right)^7$ using the power rule for exponents. We calculated the value of $8^{35}$ using a calculator and estimated the value using the properties of exponents. The final answer is $\boxed{9.33262154439 \times 10^{35}}.$

Additional Resources

For more information on exponents and power rules, check out the following resources:

• Khan Academy: Exponents and Power Rules
• Mathway: Exponents and Power Rules
• Wolfram Alpha: Exponents and Power Rules

Frequently Asked Questions

Q: What is the power rule for exponents? A: The power rule for exponents states that when we raise a power to another power, we multiply the exponents.

Q: How do I simplify an expression with multiple exponents? A: Use the power rule for exponents to simplify the expression.

$$ $$$$

Q: What is the power rule for exponents?

A: The power rule for exponents states that when we raise a power to another power, we multiply the exponents. In other words, $(a^m)^n = a^{m \times n}.$ This rule helps us simplify expressions with multiple exponents.

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

A: To simplify an expression with multiple exponents, use the power rule for exponents. For example, if we have the expression $(2^3)^4,$ we can simplify it as follows:

$(2^3)^4 = 2^{3 \times 4} = 2^12$

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

A: An exponent is a number that is raised to a power, while a power is the result of raising a number to an exponent. For example, in the expression $2^3,$ the number 2 is the base and the exponent 3 is the power.

Q: How do I calculate the value of a large exponent?

A: To calculate the value of a large exponent, use a calculator or estimate the value using the properties of exponents. For example, if we want to calculate the value of $2^{100},$ we can estimate it as follows:

$2^{100} = (2^4)^{25} = 16^{25} \approx 10^{50}$

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

A: When multiplying exponents with the same base, we add the exponents. In other words, $a^m \times a^n = a^{m + n}.$ For example, if we have the expression $2^3 \times 2^4,$ we can simplify it as follows:

$2^3 \times 2^4 = 2^{3 + 4} = 2^7$

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

A: To simplify an expression with a negative exponent, we can rewrite it as a fraction. For example, if we have the expression $2^{-3},$ we can rewrite it as follows:

$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

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

A: When dividing exponents with the same base, we subtract the exponents. In other words, $\frac{a^m}{a^n} = a^{m - n}.$ For example, if we have the expression $\frac{2^5}{2^3},$ we can simplify it as follows:

$\frac{2^5}{2^3} = 2^{5 - 3} = 2^2 = 4$

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, if we have the expression $2^0,$ we can simplify it as follows:

$2^0 = 1$

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

A: When raising a power to another power, we multiply the exponents. In other words, $(a^m)^n = a^{m \times n}.$ For example, if we have the expression $(2^3)^4,$ we can simplify it as follows:

$(2^3)^4 = 2^{3 \times 4} = 2^{12}$

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

A: To simplify an expression with multiple bases, we can use the product rule for exponents. For example, if we have the expression $2^3 \times 3^4,$ we can simplify it as follows:

$2^3 \times 3^4 = (2 \times 3)^{3 + 4} = 6^7$

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

A: When raising a product to a power, we raise each factor to that power. In other words, $(ab)^n = a^n \times b^n.$ For example, if we have the expression $(2 \times 3)^4,$ we can simplify it as follows:

$(2 \times 3)^4 = 2^4 \times 3^4 = 16 \times 81 = 1296$

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 root. For example, if we have the expression $2^{\frac{1}{2}},$ we can rewrite it as follows:

$2^{\frac{1}{2}} = \sqrt{2}$

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

A: When raising a quotient to a power, we raise the numerator and denominator to that power. In other words, $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}.$ For example, if we have the expression $\left(\frac{2}{3}\right)^4,$ we can simplify it as follows:

$\left(\frac{2}{3}\right)^4 = \frac{2^4}{3^4} = \frac{16}{81}$