Simplify The Expression: 4 3 ⋅ 4 4 4 9 \frac{4^3 \cdot 4^4}{4^9} 4 9 4 3 ⋅ 4 4 ​ Choose The Correct Answer:A. 4 16 4^{16} 4 16 B. 4 2 4^2 4 2 C. 1 4 2 \frac{1}{4^2} 4 2 1 ​ D. 1 4 \frac{1}{4} 4 1 ​ E. 0

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Introduction

In this article, we will simplify the given expression 43⋅4449\frac{4^3 \cdot 4^4}{4^9} using the properties of exponents. We will explore the rules of exponents, including the product rule, quotient rule, and power rule, to simplify the expression.

Understanding Exponents

Exponents are a shorthand way of writing repeated multiplication. For example, 434^3 can be written as 4⋅4⋅44 \cdot 4 \cdot 4. Exponents are used to represent the power or the exponentiation of a number. In this case, 434^3 means 44 raised to the power of 33.

Properties of Exponents

There are three main properties of exponents that we will use to simplify the expression:

  • Product Rule: am⋅an=am+na^m \cdot a^n = a^{m+n}
  • Quotient Rule: aman=am−n\frac{a^m}{a^n} = a^{m-n}
  • Power Rule: (am)n=amn(a^m)^n = a^{mn}

Simplifying the Expression

Using the product rule, we can simplify the numerator of the expression:

43⋅44=43+4=474^3 \cdot 4^4 = 4^{3+4} = 4^7

Now, we can rewrite the original expression as:

4749\frac{4^7}{4^9}

Using the quotient rule, we can simplify the expression:

4749=47−9=4−2\frac{4^7}{4^9} = 4^{7-9} = 4^{-2}

Reciprocal of a Negative Exponent

The reciprocal of a negative exponent is equal to the reciprocal of the base raised to the positive exponent. In this case, 4−24^{-2} is equal to 142\frac{1}{4^2}.

Conclusion

In conclusion, the simplified expression is 142\frac{1}{4^2}. This is the correct answer.

Answer

The correct answer is:

  • C. 142\frac{1}{4^2}

Explanation

The expression 43⋅4449\frac{4^3 \cdot 4^4}{4^9} can be simplified using the properties of exponents. We used the product rule to simplify the numerator, and the quotient rule to simplify the expression. The final answer is 142\frac{1}{4^2}.

Final Answer

The final answer is 142\boxed{\frac{1}{4^2}}.

Additional Information

  • Exponent Rules: The product rule, quotient rule, and power rule are essential rules to simplify expressions with exponents.
  • Negative Exponents: Negative exponents can be rewritten as the reciprocal of the base raised to the positive exponent.
  • Simplifying Expressions: Simplifying expressions with exponents requires a clear understanding of the properties of exponents.
    Simplify the Expression: 43⋅4449\frac{4^3 \cdot 4^4}{4^9} Q&A =====================================================

Introduction

In our previous article, we simplified the expression 43⋅4449\frac{4^3 \cdot 4^4}{4^9} using the properties of exponents. In this article, we will answer some frequently asked questions related to simplifying expressions with exponents.

Q&A

Q: What is the product rule of exponents?

A: The product rule of exponents states that when we multiply two numbers with the same base, we add their exponents. In other words, am⋅an=am+na^m \cdot a^n = a^{m+n}.

Q: What is the quotient rule of exponents?

A: The quotient rule of exponents states that when we divide two numbers with the same base, we subtract their exponents. In other words, aman=am−n\frac{a^m}{a^n} = a^{m-n}.

Q: What is the power rule of exponents?

A: The power rule of exponents states that when we raise a number with an exponent to another power, we multiply the exponents. In other words, (am)n=amn(a^m)^n = a^{mn}.

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

A: To simplify an expression with a negative exponent, we can rewrite it as the reciprocal of the base raised to the positive exponent. In other words, a−m=1ama^{-m} = \frac{1}{a^m}.

Q: What is the rule for simplifying expressions with the same base?

A: When we have an expression with the same base, we can simplify it by adding or subtracting the exponents, depending on whether we are multiplying or dividing.

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

A: Yes, you can simplify an expression with a variable exponent by using the properties of exponents. For example, if we have the expression x2+3x^{2+3}, we can simplify it by adding the exponents: x2+3=x5x^{2+3} = x^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 can rewrite it as the reciprocal of the base raised to the positive exponent. In other words, amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}.

Q: What is the rule for simplifying expressions with exponents and fractions?

A: When we have an expression with exponents and fractions, we can simplify it by using the properties of exponents and fractions. For example, if we have the expression x2x3\frac{x^2}{x^3}, we can simplify it by subtracting the exponents: x2x3=x2−3=x−1=1x\frac{x^2}{x^3} = x^{2-3} = x^{-1} = \frac{1}{x}.

Conclusion

In conclusion, simplifying expressions with exponents requires a clear understanding of the properties of exponents. By using the product rule, quotient rule, and power rule, we can simplify expressions with exponents and fractions. We hope this article has helped you understand how to simplify expressions with exponents.

Additional Information

  • Exponent Rules: The product rule, quotient rule, and power rule are essential rules to simplify expressions with exponents.
  • Negative Exponents: Negative exponents can be rewritten as the reciprocal of the base raised to the positive exponent.
  • Simplifying Expressions: Simplifying expressions with exponents requires a clear understanding of the properties of exponents.

Final Answer

The final answer is 142\boxed{\frac{1}{4^2}}.