Simplify The Expression: (b) 9 A ⋅ 4 0 − 1 9^a \cdot 4^{0-1} 9 A ⋅ 4 0 − 1

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Introduction

In this article, we will simplify the given expression $9^a \cdot 4^{0-1}$. This involves applying the rules of exponents and simplifying the expression to its simplest form. We will use the properties of exponents, such as the product of powers rule and the power of a power rule, to simplify the expression.

Understanding the Expression

The given expression is $9^a \cdot 4^{0-1}$. To simplify this expression, we need to understand the properties of exponents. The expression $9^a$ represents $9$ raised to the power of $a$, and $4^{0-1}$ represents $4$ raised to the power of $-1$. We will use the properties of exponents to simplify these expressions.

Simplifying the Expression

To simplify the expression $9^a \cdot 4^{0-1}$, we will use the product of powers rule, which states that when we multiply two powers with the same base, we add the exponents. In this case, we have two powers with the same base, $9$ and $4$, but with different exponents.

First, let's simplify the expression $4^{0-1}$. We can rewrite $0-1$ as $-1$, so the expression becomes $4^{-1}$. This represents $4$ raised to the power of $-1$.

Now, let's simplify the expression $9^a \cdot 4^{-1}$. We can use the product of powers rule to simplify this expression. When we multiply two powers with the same base, we add the exponents. In this case, we have two powers with the same base, $9$ and $4$, but with different exponents.

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

$$9^a \cdot 4^{-1} = \frac{9^a}{4^1}$$

This represents the expression $9^a$ divided by $4^1$.

Simplifying the Fraction

Now, let's simplify the fraction $\frac{9^a}{4^1}$. We can rewrite $4^1$ as $4$, so the fraction becomes $\frac{9^a}{4}$.

To simplify this fraction, we can use the fact that $9$ can be written as $3^2$. This allows us to rewrite the fraction as follows:

$$\frac{9^a}{4} = \frac{(3^2)^a}{4}$$

Using the power of a power rule, we can simplify this expression as follows:

$$(3^2)^a = 3^{2a}$$

So, the expression becomes:

$$\frac{3^{2a}}{4}$$

Simplifying the Expression Further

Now, let's simplify the expression $\frac{3^{2a}}{4}$. We can rewrite $4$ as $2^2$, so the expression becomes:

$$\frac{3^{2a}}{2^2}$$

To simplify this expression, we can use the fact that $3$ and $2$ are relatively prime. This allows us to rewrite the expression as follows:

$$\frac{3^{2a}}{2^2} = \frac{3^{2a}}{2^2} \cdot \frac{2^{-2a}}{2^{-2a}}$$

Using the product of powers rule, we can simplify this expression as follows:

$$\frac{3^{2a}}{2^2} \cdot \frac{2^{-2a}}{2^{-2a}} = \frac{3^{2a} \cdot 2^{-2a}}{2^2 \cdot 2^{-2a}}$$

Simplifying the numerator and denominator, we get:

$$\frac{3^{2a} \cdot 2^{-2a}}{2^2 \cdot 2^{-2a}} = \frac{3^{2a}}{2^2}$$

Conclusion

In this article, we simplified the expression $9^a \cdot 4^{0-1}$. We used the product of powers rule and the power of a power rule to simplify the expression. We also used the fact that $9$ can be written as $3^2$ and that $3$ and $2$ are relatively prime. The simplified expression is $\frac{3^{2a}}{2^2}$.

Final Answer

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Introduction

In our previous article, we simplified the expression $9^a \cdot 4^{0-1}$. In this article, we will answer some common questions related to the simplification of this expression.

Q: What is the product of powers rule?

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

Q: How do we simplify the expression $4^{0-1}$?

A: To simplify the expression $4^{0-1}$, we can rewrite $0-1$ as $-1$, so the expression becomes $4^{-1}$. This represents $4$ raised to the power of $-1$.

Q: How do we simplify the expression $9^a \cdot 4^{-1}$?

A: We can use the product of powers rule to simplify this expression. When we multiply two powers with the same base, we add the exponents. In this case, we have two powers with the same base, $9$ and $4$, but with different exponents.

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

$$9^a \cdot 4^{-1} = \frac{9^a}{4^1}$$

Q: How do we simplify the fraction $\frac{9^a}{4^1}$?

A: We can rewrite $4^1$ as $4$, so the fraction becomes $\frac{9^a}{4}$. To simplify this fraction, we can use the fact that $9$ can be written as $3^2$. This allows us to rewrite the fraction as follows:

$$\frac{9^a}{4} = \frac{(3^2)^a}{4}$$

Using the power of a power rule, we can simplify this expression as follows:

$$(3^2)^a = 3^{2a}$$

So, the expression becomes:

$$\frac{3^{2a}}{4}$$

Q: How do we simplify the expression $\frac{3^{2a}}{4}$ further?

A: We can rewrite $4$ as $2^2$, so the expression becomes:

$$\frac{3^{2a}}{2^2}$$

To simplify this expression, we can use the fact that $3$ and $2$ are relatively prime. This allows us to rewrite the expression as follows:

$$\frac{3^{2a}}{2^2} = \frac{3^{2a}}{2^2} \cdot \frac{2^{-2a}}{2^{-2a}}$$

Using the product of powers rule, we can simplify this expression as follows:

$$\frac{3^{2a}}{2^2} \cdot \frac{2^{-2a}}{2^{-2a}} = \frac{3^{2a} \cdot 2^{-2a}}{2^2 \cdot 2^{-2a}}$$

Simplifying the numerator and denominator, we get:

$$\frac{3^{2a} \cdot 2^{-2a}}{2^2 \cdot 2^{-2a}} = \frac{3^{2a}}{2^2}$$

Q: What is the final answer?

A: The final answer is $\boxed{\frac{3^{2a}}{2^2}}$.

Conclusion

In this article, we answered some common questions related to the simplification of the expression $9^a \cdot 4^{0-1}$. We used the product of powers rule and the power of a power rule to simplify the expression. We also used the fact that $9$ can be written as $3^2$ and that $3$ and $2$ are relatively prime. The simplified expression is $\frac{3^{2a}}{2^2}$.

Final Answer

The final answer is $\boxed{\frac{3^{2a}}{2^2}}$.