Work Out The Value Of \left(8^2 \times 8\right) \div \left(8^9 \div 8^5\right ]. Give Your Answer As A Decimal.

Introduction

In mathematics, exponential expressions are a fundamental concept that can be used to represent various real-world situations. When dealing with exponential expressions, it's essential to understand the rules of exponents and how to simplify them. In this article, we will explore how to work out the value of a given exponential expression, $\left(8^2 \times 8\right) \div \left(8^9 \div 8^5\right)$, and provide a step-by-step guide on how to simplify it.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $8^2$ can be read as "8 to the power of 2" or "8 squared." It means that 8 is multiplied by itself 2 times, which equals 64. Similarly, $8^9$ means that 8 is multiplied by itself 9 times.

Simplifying Exponential Expressions

To simplify 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.

Step 1: Simplify the Expression Inside the Parentheses

Let's start by simplifying the expression inside the parentheses: $\left(8^2 \times 8\right)$. We can rewrite this expression as $8^2 \times 8^1$, since $8^1$ is equal to 8.

# Step 1: Simplify the Expression Inside the Parentheses
## Expression: (8^2 × 8)
## Simplified Expression: 8^2 × 8^1

Step 2: Apply the Product of Powers Rule

The product of powers rule states that when multiplying two exponential expressions with the same base, we can add their exponents. In this case, we have $8^2 \times 8^1$, which can be simplified to $8^{2+1}$.

# Step 2: Apply the Product of Powers Rule
## Expression: 8^2 × 8^1
## Simplified Expression: 8^(2+1)
## Simplified Expression: 8^3

Step 3: Simplify the Expression Inside the Parentheses

Now, let's simplify the expression inside the parentheses: $\left(8^9 \div 8^5\right)$. We can rewrite this expression as $8^{9-5}$, since the division of two exponential expressions with the same base can be simplified by subtracting their exponents.

# Step 3: Simplify the Expression Inside the Parentheses
## Expression: (8^9 ÷ 8^5)
## Simplified Expression: 8^(9-5)
## Simplified Expression: 8^4

Step 4: Simplify the Original Expression

Now that we have simplified the expressions inside the parentheses, we can rewrite the original expression as $\frac{8^3}{8^4}$.

# Step 4: Simplify the Original Expression
## Expression: (8^2 × 8) ÷ (8^9 ÷ 8^5)
## Simplified Expression: 8^3 ÷ 8^4

Step 5: Apply the Quotient of Powers Rule

The quotient of powers rule states that when dividing two exponential expressions with the same base, we can subtract their exponents. In this case, we have $\frac{8^3}{8^4}$, which can be simplified to $8^{3-4}$.

# Step 5: Apply the Quotient of Powers Rule
## Expression: 8^3 ÷ 8^4
## Simplified Expression: 8^(3-4)
## Simplified Expression: 8^(-1)

Step 6: Simplify the Negative Exponent

A negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base. In this case, $8^{-1}$ can be rewritten as $\frac{1}{8}$.

# Step 6: Simplify the Negative Exponent
## Expression: 8^(-1)
## Simplified Expression: 1/8

Conclusion

In this article, we have explored how to work out the value of a given exponential expression, $\left(8^2 \times 8\right) \div \left(8^9 \div 8^5\right)$. We have applied the product of powers rule, the quotient of powers rule, and simplified negative exponents to arrive at the final answer. The final answer is $\boxed{0.125}$.

Final Answer

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Q: What is the order of operations for exponential expressions?

A: The order of operations for exponential expressions is:

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: How do I simplify an exponential expression?

A: To simplify 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 product of powers rule?

A: The product of powers rule states that when multiplying two exponential expressions with the same base, you can add their exponents. For example, $a^m \times a^n = a^{m+n}$.

Q: What is the quotient of powers rule?

A: The quotient of powers rule states that when dividing two exponential expressions with the same base, you can subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Q: How do I simplify a negative exponent?

A: A negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base. For example, $a^{-m} = \frac{1}{a^m}$.

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

A: An exponential expression is an expression that contains a base raised to a power, such as $a^m$. A polynomial expression is an expression that contains variables and coefficients, such as $ax^2 + bx + c$.

Q: Can I simplify an exponential expression with a variable base?

A: Yes, you can simplify an exponential expression with a variable base by following the same rules as before. For example, if you have the expression $x^2 \times x^3$, you can simplify it by adding the exponents: $x^{2+3} = x^5$.

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

A: To evaluate an exponential expression with a negative exponent, you need to take the reciprocal of the base and change the sign of the exponent. For example, if you have the expression $a^{-m}$, you can evaluate it by taking the reciprocal of $a$ and changing the sign of the exponent: $\frac{1}{a^m}$.

Q: Can I simplify an exponential expression with a fractional exponent?

A: Yes, you can simplify an exponential expression with a fractional exponent by following the same rules as before. For example, if you have the expression $a^{\frac{m}{n}}$, you can simplify it by raising the base to the power of the numerator and taking the nth root of the result.

Conclusion

In this article, we have answered some of the most frequently asked questions about exponential expressions. We have covered topics such as the order of operations, simplifying exponential expressions, the product of powers rule, the quotient of powers rule, and more. We hope that this article has been helpful in clarifying any confusion you may have had about exponential expressions.