Apply All Relevant Properties Of Exponents To Simplify The Following Expression Completely.\[$\frac{5}{11} Q B^7 \cdot 55 Q^9 B^2\$\]

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Introduction

Exponents are a fundamental concept in mathematics, and understanding how to apply their properties is crucial for simplifying complex expressions. In this article, we will explore the relevant properties of exponents and apply them to simplify the given expression: 511qb7⋅55q9b2\frac{5}{11} q b^7 \cdot 55 q^9 b^2. By the end of this guide, you will be able to simplify exponential expressions with ease.

Properties of Exponents

Before we dive into simplifying the given expression, let's review the relevant properties of exponents:

  • Product of Powers Property: When multiplying two powers with the same base, add the exponents. For example, am⋅an=am+na^m \cdot a^n = a^{m+n}.
  • Power of a Power Property: When raising a power to another power, multiply the exponents. For example, (am)n=am⋅n(a^m)^n = a^{m \cdot n}.
  • Quotient of Powers Property: When dividing two powers with the same base, subtract the exponents. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}.
  • Zero Exponent Property: Any non-zero number raised to the power of zero is equal to 1. For example, a0=1a^0 = 1.
  • Negative Exponent Property: A negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base. For example, a−m=1ama^{-m} = \frac{1}{a^m}.

Simplifying the Given Expression

Now that we have reviewed the relevant properties of exponents, let's apply them to simplify the given expression: 511qb7⋅55q9b2\frac{5}{11} q b^7 \cdot 55 q^9 b^2.

Step 1: Simplify the Coefficients

The first step is to simplify the coefficients of the expression. We can start by simplifying the fraction 511\frac{5}{11}.

$\frac{5}{11} = \frac{5}{11}$

Next, we can simplify the coefficient 55.

$55 = 5 \cdot 11$

Now, we can substitute the simplified coefficients back into the expression.

$\frac{5}{11} q b^7 \cdot 55 q^9 b^2 = \frac{5}{11} q b^7 \cdot 5 \cdot 11 q^9 b^2$

Step 2: Apply the Product of Powers Property

Now that we have simplified the coefficients, we can apply the product of powers property to simplify the expression.

$\frac{5}{11} q b^7 \cdot 5 \cdot 11 q^9 b^2 = \frac{5}{11} \cdot 5 \cdot 11 \cdot q \cdot q^9 \cdot b^7 \cdot b^2$

Next, we can combine the like terms.

$\frac{5}{11} \cdot 5 \cdot 11 \cdot q \cdot q^9 \cdot b^7 \cdot b^2 = \frac{5}{11} \cdot 5 \cdot 11 \cdot q^{1+9} \cdot b^{7+2}$

Step 3: Simplify the Exponents

Now that we have combined the like terms, we can simplify the exponents.

$\frac{5}{11} \cdot 5 \cdot 11 \cdot q^{1+9} \cdot b^{7+2} = \frac{5}{11} \cdot 5 \cdot 11 \cdot q^{10} \cdot b^9$

Step 4: Apply the Zero Exponent Property

Finally, we can apply the zero exponent property to simplify the expression.

$\frac{5}{11} \cdot 5 \cdot 11 \cdot q^{10} \cdot b^9 = \frac{5}{11} \cdot 5 \cdot 11 \cdot q^{10} \cdot b^9 \cdot 1$

Now, we can substitute the value of 1 back into the expression.

$\frac{5}{11} \cdot 5 \cdot 11 \cdot q^{10} \cdot b^9 \cdot 1 = \frac{5}{11} \cdot 5 \cdot 11 \cdot q^{10} \cdot b^9$

Step 5: Simplify the Expression

Now that we have applied all the relevant properties of exponents, we can simplify the expression.

$\frac{5}{11} \cdot 5 \cdot 11 \cdot q^{10} \cdot b^9 = 5^2 \cdot 11^2 \cdot q^{10} \cdot b^9$

Finally, we can simplify the expression by combining the like terms.

$5^2 \cdot 11^2 \cdot q^{10} \cdot b^9 = 25 \cdot 121 \cdot q^{10} \cdot b^9$

Step 6: Simplify the Expression Further

Now that we have simplified the expression, we can simplify it further by combining the like terms.

$25 \cdot 121 \cdot q^{10} \cdot b^9 = 3025 \cdot q^{10} \cdot b^9$

Step 7: Simplify the Expression Even Further

Finally, we can simplify the expression even further by combining the like terms.

$3025 \cdot q^{10} \cdot b^9 = 3025 q^{10} b^9$

Conclusion

In this article, we have applied the relevant properties of exponents to simplify the given expression: 511qb7⋅55q9b2\frac{5}{11} q b^7 \cdot 55 q^9 b^2. By following the step-by-step guide, we have simplified the expression to its final form: 3025q10b93025 q^{10} b^9. We hope that this guide has been helpful in understanding how to apply the properties of exponents to simplify complex expressions.

Final Answer

Introduction

In our previous article, we explored the relevant properties of exponents and applied them to simplify the expression: 511qb7⋅55q9b2\frac{5}{11} q b^7 \cdot 55 q^9 b^2. In this article, we will answer some frequently asked questions about simplifying exponential expressions.

Q: What are the most common properties of exponents?

A: The most common properties of exponents are:

  • Product of Powers Property: When multiplying two powers with the same base, add the exponents. For example, am⋅an=am+na^m \cdot a^n = a^{m+n}.
  • Power of a Power Property: When raising a power to another power, multiply the exponents. For example, (am)n=am⋅n(a^m)^n = a^{m \cdot n}.
  • Quotient of Powers Property: When dividing two powers with the same base, subtract the exponents. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}.
  • Zero Exponent Property: Any non-zero number raised to the power of zero is equal to 1. For example, a0=1a^0 = 1.
  • Negative Exponent Property: A negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base. For example, a−m=1ama^{-m} = \frac{1}{a^m}.

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

A: To simplify an expression with multiple bases, you need to apply the product of powers property. For example, if you have the expression am⋅bna^m \cdot b^n, you can simplify it by adding the exponents: am⋅bn=am⋅bn=am+na^m \cdot b^n = a^m \cdot b^n = a^{m+n}.

Q: What if I have an expression with a negative exponent?

A: If you have an expression with a negative exponent, you can rewrite it as a positive exponent by taking the reciprocal of the base. For example, if you have the expression a−ma^{-m}, you can rewrite it as 1am\frac{1}{a^m}.

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

A: Yes, you can simplify an expression with a zero exponent. According to the zero exponent property, any non-zero number raised to the power of zero is equal to 1. For example, a0=1a^0 = 1.

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

A: To simplify an expression with multiple terms, you need to apply the product of powers property and the quotient of powers property. For example, if you have the expression am⋅bn⋅cpa^m \cdot b^n \cdot c^p, you can simplify it by adding the exponents: am⋅bn⋅cp=am+n+pa^m \cdot b^n \cdot c^p = a^{m+n+p}.

Q: What if I have an expression with a variable base?

A: If you have an expression with a variable base, you need to apply the product of powers property and the power of a power property. For example, if you have the expression (am)n(a^m)^n, you can simplify it by multiplying the exponents: (am)n=am⋅n(a^m)^n = a^{m \cdot n}.

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

A: Yes, you can simplify an expression with a fractional exponent. According to the quotient of powers property, when dividing two powers with the same base, subtract the exponents. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}.

Conclusion

In this article, we have answered some frequently asked questions about simplifying exponential expressions. We hope that this guide has been helpful in understanding how to apply the properties of exponents to simplify complex expressions.

Final Answer

The final answer is: There is no final numerical answer to this problem. The goal of this article is to provide a guide on how to simplify exponential expressions.