Simplify ( 5 2 A − 4 B 7 ) − 3 \left(\frac{5}{2} A^{-4} B^7\right)^{-3} ( 2 5 ​ A − 4 B 7 ) − 3 .A. 125 B 21 8 A 12 \frac{125 B^{21}}{8 A^{12}} 8 A 12 125 B 21 ​ B. 8 A 12 125 B 21 \frac{8 A^{12}}{125 B^{21}} 125 B 21 8 A 12 ​ C. 125 A 12 8 B 21 \frac{125 A^{12}}{8 B^{21}} 8 B 21 125 A 12 ​ D. 8 B 4 125 A 7 \frac{8 B^4}{125 A^7} 125 A 7 8 B 4 ​

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Understanding Exponents and Their Rules

Exponents are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will focus on simplifying the expression (52a−4b7)−3\left(\frac{5}{2} a^{-4} b^7\right)^{-3} using the rules of exponents.

The Rules of Exponents

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

  • Product of Powers Rule: When multiplying two powers with the same base, add their exponents. For example, am⋅an=am+na^m \cdot a^n = a^{m+n}.
  • Power of a Power Rule: 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 Rule: When dividing two powers with the same base, subtract their exponents. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}.
  • Negative Exponent Rule: 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 rules of exponents, let's apply them to simplify the given expression:

(52a−4b7)−3\left(\frac{5}{2} a^{-4} b^7\right)^{-3}

Using the Power of a Power Rule, we can rewrite the expression as:

52a−4⋅−3b7⋅−3\frac{5}{2} a^{-4 \cdot -3} b^{7 \cdot -3}

Simplifying the exponents, we get:

52a12b−21\frac{5}{2} a^{12} b^{-21}

Using the Negative Exponent Rule, we can rewrite the expression as:

52a12b21\frac{5}{2} \frac{a^{12}}{b^{21}}

Now, let's simplify the fraction by multiplying the numerator and denominator by the reciprocal of the denominator:

5a122b21\frac{5 a^{12}}{2 b^{21}}

Comparing the Simplified Expression with the Answer Choices

Now that we have simplified the expression, let's compare it with the answer choices:

A. 125b218a12\frac{125 b^{21}}{8 a^{12}} B. 8a12125b21\frac{8 a^{12}}{125 b^{21}} C. 125a128b21\frac{125 a^{12}}{8 b^{21}} D. 8b4125a7\frac{8 b^4}{125 a^7}

Comparing the simplified expression with the answer choices, we can see that the correct answer is:

C. 125a128b21\frac{125 a^{12}}{8 b^{21}}

Conclusion

In this article, we have simplified the expression (52a−4b7)−3\left(\frac{5}{2} a^{-4} b^7\right)^{-3} using the rules of exponents. We have applied the Power of a Power Rule, Negative Exponent Rule, and Quotient of Powers Rule to simplify the expression. Finally, we have compared the simplified expression with the answer choices and determined that the correct answer is C. 125a128b21\frac{125 a^{12}}{8 b^{21}}.

Practice Problems

To practice simplifying exponents, try the following problems:

  1. Simplify the expression (34x5y−2)−2\left(\frac{3}{4} x^5 y^{-2}\right)^{-2}.
  2. Simplify the expression (23a3b−4)−1\left(\frac{2}{3} a^3 b^{-4}\right)^{-1}.
  3. Simplify the expression (56c2d3)−3\left(\frac{5}{6} c^2 d^3\right)^{-3}.

Answer Key

  1. 16x5y49\frac{16 x^5 y^4}{9}
  2. 3a3b42\frac{3 a^3 b^4}{2}
  3. 6c6d95\frac{6 c^6 d^9}{5}
    Frequently Asked Questions: Simplifying Exponents =====================================================

Q: What is the rule for simplifying exponents when the base is the same?

A: When the base is the same, you can add the exponents. For example, am⋅an=am+na^m \cdot a^n = a^{m+n}.

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

A: To simplify an expression with a negative exponent, you can rewrite it as a positive exponent by taking the reciprocal of the base. For example, a−m=1ama^{-m} = \frac{1}{a^m}.

Q: What is the rule for simplifying exponents when the base is different?

A: When the base is different, you can multiply the exponents. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}.

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

A: To simplify an expression with multiple exponents, you can apply the rules of exponents in the following order:

  1. Simplify any negative exponents by rewriting them as positive exponents.
  2. Apply the power of a power rule to simplify any expressions with multiple exponents.
  3. Apply the product of powers rule to simplify any expressions with multiple bases.

Q: What is the difference between a power and an exponent?

A: A power is the result of raising a base to a certain exponent. For example, ama^m is a power. An exponent is the number that is raised to a certain power. For example, mm is an exponent.

Q: How do you simplify an expression with a fractional exponent?

A: To simplify an expression with a fractional exponent, you can rewrite it as a product of two exponents. For example, amn=(am)1na^{\frac{m}{n}} = (a^m)^{\frac{1}{n}}.

Q: What is the rule for simplifying exponents when the base is a fraction?

A: When the base is a fraction, you can simplify the expression by rewriting it as a product of two fractions. For example, (ab)m=ambm\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}.

Q: How do you simplify an expression with multiple bases and exponents?

A: To simplify an expression with multiple bases and exponents, you can apply the rules of exponents in the following order:

  1. Simplify any negative exponents by rewriting them as positive exponents.
  2. Apply the power of a power rule to simplify any expressions with multiple exponents.
  3. Apply the product of powers rule to simplify any expressions with multiple bases.
  4. Simplify any fractions by rewriting them as products of two fractions.

Q: What is the difference between a radical and an exponent?

A: A radical is a symbol that represents a root of a number. For example, a\sqrt{a} is a radical. An exponent is a number that is raised to a certain power. For example, ama^m is an exponent.

Q: How do you simplify an expression with a radical and an exponent?

A: To simplify an expression with a radical and an exponent, you can apply the rules of exponents and radicals in the following order:

  1. Simplify any negative exponents by rewriting them as positive exponents.
  2. Apply the power of a power rule to simplify any expressions with multiple exponents.
  3. Apply the product of powers rule to simplify any expressions with multiple bases.
  4. Simplify any radicals by rewriting them as products of two radicals.

Conclusion

In this article, we have answered some frequently asked questions about simplifying exponents. We have covered topics such as simplifying expressions with negative exponents, multiple exponents, and fractional exponents. We have also discussed the rules for simplifying exponents when the base is the same, different, or a fraction. Finally, we have provided some examples of how to simplify expressions with radicals and exponents.