Simplify The Expression:h) 2 ( A 4 ) 2 2\left(a^4\right)^2 2 ( A 4 ) 2

Understanding Exponents and Simplification

In mathematics, exponents are a shorthand way of representing repeated multiplication of a number. When we see an expression like $a^4$, it means that the number $a$ is multiplied by itself four times. For example, $a^4 = a \times a \times a \times a$. Exponents can be simplified using the rules of multiplication and exponentiation.

Simplifying the Expression

The given expression is $2\left(a^4\right)^2$. To simplify this expression, we need to apply the rules of exponentiation. When we have an exponent raised to another exponent, we multiply the exponents. In this case, we have $\left(a^4\right)^2$, which means that $a^4$ is multiplied by itself two times.

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

$$2\left(a^4\right)^2 = 2 \times a^{4 \times 2} = 2 \times a^8$$

Breaking Down the Simplification

Let's break down the simplification step by step:

1. Applying the rule of exponentiation: When we have an exponent raised to another exponent, we multiply the exponents. In this case, we have $\left(a^4\right)^2$, which means that $a^4$ is multiplied by itself two times.
2. Simplifying the exponent: When we multiply the exponents, we get $4 \times 2 = 8$. So, the expression becomes $2 \times a^8$.
3. Multiplying the coefficients: The coefficient of the expression is 2, and we have another 2 from the exponentiation. Multiplying these two coefficients, we get $2 \times 2 = 4$.

The Final Simplified Expression

After applying the rules of exponentiation and simplification, we get the final simplified expression:

$$2\left(a^4\right)^2 = 4a^8$$

Conclusion

In this article, we simplified the expression $2\left(a^4\right)^2$ using the rules of exponentiation and simplification. We applied the rule of exponentiation to multiply the exponents and then simplified the resulting expression. The final simplified expression is $4a^8$.

Common Mistakes to Avoid

When simplifying expressions with exponents, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not applying the rule of exponentiation: When we have an exponent raised to another exponent, we must multiply the exponents.
• Not simplifying the exponent: When we multiply the exponents, we must simplify the resulting exponent.
• Not multiplying the coefficients: When we have multiple coefficients, we must multiply them together.

Practice Problems

To practice simplifying expressions with exponents, try the following problems:

1. Simplify the expression $3\left(b^3\right)^2$.
2. Simplify the expression $2\left(c^2\right)^3$.
3. Simplify the expression $4\left(d^5\right)^2$.

Answer Key

1. $9b^6$
2. $8c^6$
3. $16d^{10}$
   Frequently Asked Questions: Simplifying Expressions with Exponents ====================================================================

Q: What is the rule of exponentiation?

A: The rule of exponentiation states that when we have an exponent raised to another exponent, we multiply the exponents. For example, $\left(a^4\right)^2 = a^{4 \times 2} = a^8$.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, follow these steps:

1. Apply the rule of exponentiation: When we have an exponent raised to another exponent, we multiply the exponents.
2. Simplify the exponent: When we multiply the exponents, we must simplify the resulting exponent.
3. Multiply the coefficients: When we have multiple coefficients, we must multiply them together.

Q: What is the difference between $a^4$ and $\left(a^4\right)^2$?

A: $a^4$ means that the number $a$ is multiplied by itself four times, resulting in $a \times a \times a \times a$. On the other hand, $\left(a^4\right)^2$ means that $a^4$ is multiplied by itself two times, resulting in $a^8$.

Q: Can I simplify an expression with a negative exponent?

A: Yes, you can simplify an expression with a negative exponent. For example, $a^{-4} = \frac{1}{a^4}$.

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

A: An expression with a zero exponent is equal to 1. For example, $a^0 = 1$.

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

A: Yes, you can simplify an expression with a fractional exponent. For example, $a^{\frac{1}{2}} = \sqrt{a}$.

Q: What is the order of operations when simplifying expressions with exponents?

A: When simplifying expressions with exponents, follow the order of operations:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate exponents next.
3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate addition and subtraction operations from left to right.

Q: Can I simplify an expression with multiple exponents?

A: Yes, you can simplify an expression with multiple exponents. For example, $a^4 \times b^3 = a^4 \times b^3$.

Q: How do I simplify an expression with a variable in the exponent?

A: To simplify an expression with a variable in the exponent, follow the same rules as before. For example, $a^{2x} = a^{2x}$.

Conclusion

In this article, we answered frequently asked questions about simplifying expressions with exponents. We covered topics such as the rule of exponentiation, simplifying expressions with negative exponents, and the order of operations. We also provided examples and practice problems to help you master the skills of simplifying expressions with exponents.

Practice Problems

To practice simplifying expressions with exponents, try the following problems:

1. Simplify the expression $3\left(b^3\right)^2$.
2. Simplify the expression $2\left(c^2\right)^3$.
3. Simplify the expression $4\left(d^5\right)^2$.

Answer Key

1. $9b^6$
2. $8c^6$
3. $16d^{10}$