Simplify The Following Expressions: 3 18 ÷ 3 15 3^{18} \div 3^{15} 3 18 ÷ 3 15 7 0 − 10 1 7^0 - 10^1 7 0 − 1 0 1 ( 32 A B C ) 0 (32abc)^0 ( 32 Ab C ) 0 ( 3 X 2 ) 3 (3x^2)^3 ( 3 X 2 ) 3 ( 4 X Y Z 2 ) 4 ( X 0 Z ) 2 \frac{(4xyz^2)^4}{(x^0z)^2} ( X 0 Z ) 2 ( 4 X Y Z 2 ) 4 ​

Introduction

In mathematics, simplifying expressions is an essential skill that helps us to solve problems more efficiently and accurately. It involves applying various mathematical rules and properties to reduce complex expressions into simpler ones. In this article, we will simplify five given expressions using the rules of exponents and other mathematical properties.

Simplifying Expression 1: $3^{18} \div 3^{15}$

To simplify the expression $3^{18} \div 3^{15}$, we can use the quotient rule of exponents, which states that when we divide two powers with the same base, we subtract the exponents. Therefore, we can rewrite the expression as:

$$3^{18} \div 3^{15} = 3^{18-15} = 3^3$$

Using the rule that $a^m = a \cdot a \cdot \ldots \cdot a$ (m times), we can simplify $3^3$ as:

$$3^3 = 3 \cdot 3 \cdot 3 = 27$$

Therefore, the simplified expression is $27$.

Simplifying Expression 2: $7^0 - 10^1$

To simplify the expression $7^0 - 10^1$, we can use the rule that any number raised to the power of 0 is equal to 1. Therefore, we can rewrite the expression as:

$$7^0 - 10^1 = 1 - 10 = -9$$

Therefore, the simplified expression is $-9$.

Simplifying Expression 3: $(32abc)^0$

To simplify the expression $(32abc)^0$, we can use the rule that any number raised to the power of 0 is equal to 1. Therefore, we can rewrite the expression as:

$$(32abc)^0 = 1$$

Therefore, the simplified expression is $1$.

Simplifying Expression 4: $(3x^2)^3$

To simplify the expression $(3x^2)^3$, we can use the rule that $(ab)^m = a^m \cdot b^m$. Therefore, we can rewrite the expression as:

$$(3x^2)^3 = 3^3 \cdot (x^2)^3$$

Using the rule that $(a^m)^n = a^{mn}$, we can simplify $(x^2)^3$ as:

$$(x^2)^3 = x^{2 \cdot 3} = x^6$$

Therefore, the simplified expression is $27x^6$.

Simplifying Expression 5: $\frac{(4xyz^2)^4}{(x^0z)^2}$

To simplify the expression $\frac{(4xyz^2)^4}{(x^0z)^2}$, we can use the rule that $(ab)^m = a^m \cdot b^m$. Therefore, we can rewrite the expression as:

$$\frac{(4xyz^2)^4}{(x^0z)^2} = \frac{4^4 \cdot (xyz^2)^4}{(x^0z)^2}$$

Using the rule that $(a^m)^n = a^{mn}$, we can simplify $(xyz^2)^4$ as:

$$(xyz^2)^4 = x^4 \cdot y^4 \cdot (z^2)^4 = x^4 \cdot y^4 \cdot z^8$$

Using the rule that $(a^m)^n = a^{mn}$, we can simplify $(x^0z)^2$ as:

$$(x^0z)^2 = x^{0 \cdot 2} \cdot z^2 = z^2$$

Therefore, the simplified expression is:

$$\frac{4^4 \cdot x^4 \cdot y^4 \cdot z^8}{z^2} = 256 \cdot x^4 \cdot y^4 \cdot z^6$$

Therefore, the simplified expression is $256x^4y^4z^6$.

Conclusion

In this article, we simplified five given expressions using the rules of exponents and other mathematical properties. We used the quotient rule of exponents to simplify the first expression, the rule that any number raised to the power of 0 is equal to 1 to simplify the second and third expressions, and the rule that $(ab)^m = a^m \cdot b^m$ to simplify the fourth and fifth expressions. We also used the rule that $(a^m)^n = a^{mn}$ to simplify the fourth and fifth expressions. By applying these rules, we were able to simplify the given expressions into simpler ones.

Introduction

In our previous article, we simplified five given expressions using the rules of exponents and other mathematical properties. In this article, we will answer some frequently asked questions related to simplifying expressions.

Q: What is the quotient rule of exponents?

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

Q: What is the rule for any number raised to the power of 0?

A: Any number raised to the power of 0 is equal to 1. For example, $a^0 = 1$.

Q: What is the rule for $(ab)^m$?

A: $(ab)^m = a^m \cdot b^m$. This means that when we raise a product to a power, we raise each factor to that power.

Q: What is the rule for $(a^m)^n$?

A: $(a^m)^n = a^{mn}$. This means that when we raise a power to a power, we multiply the exponents.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can use the following steps:

1. Identify the base and the exponent.
2. Use the quotient rule of exponents to simplify the expression.
3. Use the rule that any number raised to the power of 0 is equal to 1.
4. Use the rule that $(ab)^m = a^m \cdot b^m$.
5. Use the rule that $(a^m)^n = a^{mn}$.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Not using the quotient rule of exponents when dividing powers with the same base.
• Not using the rule that any number raised to the power of 0 is equal to 1.
• Not using the rule that $(ab)^m = a^m \cdot b^m$.
• Not using the rule that $(a^m)^n = a^{mn}$.
• Not following the order of operations (PEMDAS).

Q: How do I check my work when simplifying expressions?

A: To check your work when simplifying expressions, you can use the following steps:

1. Plug in some values for the variables.
2. Simplify the expression using the rules of exponents.
3. Check that the simplified expression is equal to the original expression.
4. If the simplified expression is not equal to the original expression, go back and recheck your work.

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions has many real-world applications, including:

• Algebra: Simplifying expressions is an essential skill in algebra, where we use variables and exponents to solve equations and inequalities.
• Calculus: Simplifying expressions is also used in calculus, where we use limits and derivatives to study functions and their behavior.
• Physics: Simplifying expressions is used in physics to describe the motion of objects and the behavior of physical systems.
• Engineering: Simplifying expressions is used in engineering to design and optimize systems and processes.

Conclusion

In this article, we answered some frequently questions related to simplifying expressions. We covered the quotient rule of exponents, the rule for any number raised to the power of 0, and the rules for $(ab)^m$ and $(a^m)^n$. We also discussed some common mistakes to avoid when simplifying expressions and how to check your work. Finally, we discussed some real-world applications of simplifying expressions. By following these rules and guidelines, you can become proficient in simplifying expressions and apply this skill to a wide range of problems.