Using All Of The Laws Of Exponents, Simplify Each Of The Following:15. \left(4 A^3 B^5 C^0\right)\left(3 A^4 B^6 C^2\right ]16. \left(-3 X Y^2\right)^4\left(2 X^{10} Y^2\right ]17. $\frac{8 M^9 N^6}{4 M^2 N^{-3}}$18.

Introduction

Exponents are a fundamental concept in mathematics, and understanding the laws of exponents is crucial for simplifying complex exponential expressions. In this article, we will explore the laws of exponents and apply them to simplify four different exponential expressions.

Law of Exponents

Before we dive into the simplification of the given expressions, let's briefly review the laws of exponents. The laws of exponents are a set of rules that govern the behavior of exponents when they are multiplied, divided, or raised to a power.

• Product of Powers: When multiplying two powers with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power: When raising a power to a power, we multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Quotient of Powers: When dividing two powers with the same base, we subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.
• Zero Exponent: Any non-zero number raised to the power of zero is equal to 1. For example, $a^0 = 1$.
• Negative Exponent: A negative exponent is equivalent to taking the reciprocal of the base raised to the positive exponent. For example, $a^{-m} = \frac{1}{a^m}$.

Simplifying Expression 15

The first expression we will simplify is $\left(4 a^3 b^5 c^0\right)\left(3 a^4 b^6 c^2\right)$. To simplify this expression, we will apply the product of powers law.

$\left(4 a^3 b^5 c^0\right)\left(3 a^4 b^6 c^2\right)$
= $4 \cdot 3 \cdot a^{3+4} \cdot b^{5+6} \cdot c^{0+2}$
= $12 a^7 b^{11} c^2$

Simplifying Expression 16

The second expression we will simplify is $\left(-3 x y^2\right)^4\left(2 x^{10} y^2\right)$. To simplify this expression, we will apply the power of a power law.

$\left(-3 x y^2\right)^4\left(2 x^{10} y^2\right)$
= $(-3)^4 \cdot x^{2 \cdot 4} \cdot y^{2 \cdot 4} \cdot 2 \cdot x^{10} \cdot y^2$
= $81 \cdot x^{24} \cdot y^{12} \cdot 2 \cdot x^{10} \cdot y^2$
= $162 x^{34} y^{14}$

Simplifying Expression 17

The third expression we will simplify is $\frac{8 m^9 n^6}{4 m^2 n^{-3}}$. To simplify this expression, we will apply the quotient of powers law.

$\frac{8 m^9 n^6}{4 m^2 n^{-3}}$
= $2 \cdot m^{9-2} \cdot n^{6-(-3)}$
= $2 \cdot m^7 \cdot n^9$

Simplifying Expression 18

The fourth expression we will simplify is $\frac{16 x^8 y^4}{2 x^2 y^{-2}}$. To simplify this expression, we will apply the quotient of powers law.

$\frac{16 x^8 y^4}{2 x^2 y^{-2}}$
= $8 \cdot x^{8-2} \cdot y^{4-(-2)}$
= $8 \cdot x^6 \cdot y^6$

Conclusion

In this article, we have applied the laws of exponents to simplify four different exponential expressions. We have used the product of powers law, the power of a power law, and the quotient of powers law to simplify the expressions. By understanding and applying these laws, we can simplify complex exponential expressions and make them easier to work with.

References

• "Laws of Exponents". Khan Academy. Retrieved 2023-02-20.
• "Exponents and Powers". Math Open Reference. Retrieved 2023-02-20.

Further Reading

• "Exponents and Exponential Functions". Wolfram MathWorld. Retrieved 2023-02-20.
• "Laws of Exponents". Purplemath. Retrieved 2023-02-20.
  Frequently Asked Questions About Laws of Exponents =====================================================

Introduction

The laws of exponents are a fundamental concept in mathematics, and understanding them is crucial for simplifying complex exponential expressions. In this article, we will answer some frequently asked questions about the laws of exponents.

Q: What is the product of powers law?

A: The product of powers law states that when multiplying two powers with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.

Q: What is the power of a power law?

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

Q: What is the quotient of powers law?

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

Q: What is the zero exponent rule?

A: The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. For example, $a^0 = 1$.

Q: What is the negative exponent rule?

A: The negative exponent rule states that a negative exponent is equivalent to taking the reciprocal of the base raised to the positive exponent. For example, $a^{-m} = \frac{1}{a^m}$.

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

A: To simplify an expression with multiple exponents, you can use the laws of exponents to combine the exponents. For example, $\left(4 a^3 b^5 c^0\right)\left(3 a^4 b^6 c^2\right)$ can be simplified using the product of powers law as follows:

$\left(4 a^3 b^5 c^0\right)\left(3 a^4 b^6 c^2\right)$
= $4 \cdot 3 \cdot a^{3+4} \cdot b^{5+6} \cdot c^{0+2}$
= $12 a^7 b^{11} c^2$

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

A: To simplify an expression with a negative exponent, you can use the negative exponent rule to rewrite the expression with a positive exponent. For example, $\frac{1}{a^m}$ can be rewritten as $a^{-m}$.

Q: What are some common mistakes to avoid when working with exponents?

A: Some common mistakes to avoid when working with exponents include:

• Forgetting to apply the laws of exponents when simplifying expressions
• Not following the order of operations when simplifying expressions
• Not using the correct notation for exponents (e.g. using a caret symbol (^) instead of a superscript)

Conclusion

In this article, we have answered some frequently asked questions about the laws of exponents. We have covered topics such as the product of powers law, the power of a power law, and the quotient of powers law. By understanding and applying these laws, you can simplify complex exponential expressions and make them easier to work with.

References

• "Laws of Exponents". Khan Academy. Retrieved 2023-02-20.
• "Exponents and Powers". Math Open Reference. Retrieved 2023-02-20.

Further Reading

• "Exponents and Exponential Functions". Wolfram MathWorld. Retrieved 2023-02-20.
• "Laws of Exponents". Purplemath. Retrieved 2023-02-20.