Simplify Each Of The Following, Leaving Your Answers As A Power.1.1 { A^2 \times A^2 + A^4 $}$1.2 { 5^2 \times (3^4 - 15^6) $}$1.3 { 11^2 + (-11)^2 $}$1.4 { (5 \times 5 \times 5) \div \sqrt[3]{125} $}$1.5

Introduction

Exponents and expressions are fundamental concepts in mathematics that help us simplify complex calculations. In this article, we will explore the simplification of exponents and expressions using various mathematical operations. We will examine five different expressions and simplify each one, leaving our answers as powers.

1.1 Simplifying $a^2 \times a^2 + a^4$

The given expression is $a^2 \times a^2 + a^4$. To simplify this expression, we need to apply the rules of exponents. When we multiply two powers with the same base, we add their exponents. Therefore, $a^2 \times a^2 = a^{2+2} = a^4$. Now, we can rewrite the expression as $a^4 + a^4$. Since both terms have the same base and exponent, we can combine them by adding their coefficients. Therefore, $a^4 + a^4 = 2a^4$.

1.2 Simplifying $5^2 \times (3^4 - 15^6)$

The given expression is $5^2 \times (3^4 - 15^6)$. To simplify this expression, we need to apply the rules of exponents and order of operations. First, we evaluate the expressions inside the parentheses. $3^4 = 81$ and $15^6 = 15,625$. Now, we can rewrite the expression as $5^2 \times (81 - 15,625)$. Next, we subtract $15,625$ from $81$, which gives us $-15,544$. Now, we can rewrite the expression as $5^2 \times -15,544$. Finally, we multiply $5^2$ by $-15,544$, which gives us $-78,720$.

1.3 Simplifying $11^2 + (-11)^2$

The given expression is $11^2 + (-11)^2$. To simplify this expression, we need to apply the rules of exponents and order of operations. When we raise a negative number to an even power, the result is positive. Therefore, $(-11)^2 = 11^2 = 121$. Now, we can rewrite the expression as $121 + 121$. Since both terms have the same value, we can combine them by adding their coefficients. Therefore, $121 + 121 = 242$.

1.4 Simplifying $(5 \times 5 \times 5) \div \sqrt[3]{125}$

The given expression is $(5 \times 5 \times 5) \div \sqrt[3]{125}$. To simplify this expression, we need to apply the rules of exponents and order of operations. First, we evaluate the expression inside the parentheses. $5 \times 5 \times 5 = 125$. Now, we can rewrite the expression as $125 \div \sqrt[3]{125}$. Next, we evaluate the cube root of $125$, which gives us $5$. Now, we can rewrite the expression as $125 \div 5$. Finally, we divide $125$ by $5$, which gives us $25$.

Conclusion

In this article, we simplified five different expressions using various mathematical operations. We applied the rules of exponents, order of operations, and other mathematical concepts to simplify each expression. By following these rules and concepts, we were able to simplify each expression and leave our answers as powers. We hope that this article has provided a clear understanding of how to simplify exponents and expressions in mathematics.

Additional Tips and Resources

• To simplify exponents and expressions, it's essential to follow the order of operations (PEMDAS).
• When multiplying two powers with the same base, add their exponents.
• When raising a negative number to an even power, the result is positive.
• To evaluate the cube root of a number, divide the number by the cube root of the base.
• For more information on simplifying exponents and expressions, check out the following resources:

• Khan Academy: Exponents and Expressions
• Mathway: Exponents and Expressions
• Wolfram Alpha: Exponents and Expressions

Final Thoughts

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations 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 expression with multiple exponents?

A: To simplify an expression with multiple exponents, we need to apply the rules of exponents. When we multiply two powers with the same base, we add their exponents. For example, $a^2 \times a^3 = a^{2+3} = a^5$.

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

A: A positive exponent indicates that we are raising a number to a power, while a negative exponent indicates that we are taking the reciprocal of a number raised to a power. For example, $a^3$ is a positive exponent, while $a^{-3}$ is a negative exponent.

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

A: To simplify an expression with a negative exponent, we need to take the reciprocal of the number raised to the positive exponent. For example, $a^{-3} = \frac{1}{a^3}$.

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

A: When we multiply two powers with the same base, we add their exponents. For example, $a^2 \times a^3 = a^{2+3} = a^5$.

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

A: To simplify an expression with a fraction as an exponent, we need to apply the rules of exponents. When we raise a number to a fraction power, we can rewrite it as a root. For example, $a^{\frac{1}{2}} = \sqrt{a}$.

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

A: When we divide two powers with the same base, we subtract their exponents. For example, $a^3 \div a^2 = a^{3-2} = a^1 = a$.

Q: How do I simplify an expression with a cube root?

A: To simplify an expression with a cube root, we need to apply the rules of exponents. When we raise a number to a cube root, we can rewrite it as a fraction. For example, $\sqrt[3]{a} = a^{\frac{1}{3}}$.

Q: What is the rule for simplifying an expression with multiple terms?

A: To simplify an expression with multiple terms, we need to combine like terms. Like terms are terms that have the same variable and exponent. For example, $2a^2 + 3a^2 = 5a^2$.

Conclusion

In this article, we answered some of the most frequently asked questions about simplifying exponents and expressions. We covered topics such as the order of operations, simplifying expressions with multiple exponents, and simplifying expressions with negative exponents. We hope that this article has provided a clear understanding of how to simplify exponents and expressions in mathematics.

Additional Tips and Resources

• To simplify exponents and expressions, it's essential to follow the order of operations (PEMDAS).
• When multiplying two powers with the same base, add their exponents.
• When raising a negative number to an even power, the result is positive.
• To evaluate the cube root of a number, divide the number by the cube root of the base.
• For more information on simplifying exponents and expressions, check out the following resources:

• Khan Academy: Exponents and Expressions
• Mathway: Exponents and Expressions
• Wolfram Alpha: Exponents and Expressions

Final Thoughts

Simplifying exponents and expressions is a crucial skill in mathematics that helps us solve complex problems. By following the rules of exponents, order of operations, and other mathematical concepts, we can simplify even the most complex expressions. We hope that this article has provided a clear understanding of how to simplify exponents and expressions in mathematics.