Select The Best Answer For The Question.Simplify $\frac{3 M^{-2}}{5 N^{-3}}$A. $\frac{-4}{5 N^8}$B. $\frac{3 N^3}{5 M^2}$C. $\frac{3 M^2 N^{10}}{2}$D. $\frac{1}{m^5 N^4}$

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Understanding Exponents and Their Rules

Exponents are a fundamental concept in algebra, and understanding how to simplify expressions with exponents is crucial for solving mathematical problems. In this article, we will focus on simplifying the expression 3m−25n−3\frac{3 m^{-2}}{5 n^{-3}} and selecting the best answer from the given options.

The Rules of Exponents

Before we dive into simplifying the expression, let's review the rules of exponents. The rules state that:

  • When multiplying two numbers with the same base, we add their exponents. For example, am⋅an=am+na^m \cdot a^n = a^{m+n}.
  • When dividing two numbers with the same base, we subtract their exponents. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}.
  • When raising a power to a power, we multiply their exponents. For example, (am)n=am⋅n(a^m)^n = a^{m \cdot n}.
  • When a number is raised to a negative exponent, we can rewrite it as a fraction with the base in the denominator and the exponent as a positive number. For example, a−m=1ama^{-m} = \frac{1}{a^m}.

Simplifying the Expression

Now that we have reviewed the rules of exponents, let's simplify the expression 3m−25n−3\frac{3 m^{-2}}{5 n^{-3}}. To simplify this expression, we need to apply the rules of exponents.

First, we can rewrite the expression as 35⋅1m2⋅1n3\frac{3}{5} \cdot \frac{1}{m^2} \cdot \frac{1}{n^3}.

Next, we can apply the rule for dividing numbers with the same base, which states that we subtract their exponents. In this case, we have 1m2⋅1n3=1m2⋅n3\frac{1}{m^2} \cdot \frac{1}{n^3} = \frac{1}{m^2 \cdot n^3}.

Now, we can rewrite the expression as 35⋅1m2⋅n3\frac{3}{5} \cdot \frac{1}{m^2 \cdot n^3}.

Evaluating the Options

Now that we have simplified the expression, let's evaluate the options and select the best answer.

Option A: −45n8\frac{-4}{5 n^8}

This option is incorrect because the exponent of nn is 88, which is not equal to 33.

Option B: 3n35m2\frac{3 n^3}{5 m^2}

This option is incorrect because the exponent of mm is 22, which is not equal to −2-2.

Option C: 3m2n102\frac{3 m^2 n^{10}}{2}

This option is incorrect because the exponent of nn is 1010, which is not equal to −3-3.

Option D: 1m5n4\frac{1}{m^5 n^4}

This option is incorrect because the exponent of mm is 55, which is not equal to −2-2, and the exponent of nn is 44, which is not equal to −3-3.

The Correct Answer

After evaluating the options, we can see that none of them match the simplified expression. However, we can rewrite the expression as 35⋅1m2⋅1n3=35⋅1m2⋅n3=35⋅n3m2=3n35m2\frac{3}{5} \cdot \frac{1}{m^2} \cdot \frac{1}{n^3} = \frac{3}{5} \cdot \frac{1}{m^2 \cdot n^3} = \frac{3}{5} \cdot \frac{n^3}{m^2} = \frac{3 n^3}{5 m^2}.

Q: What is the rule for dividing numbers with the same base?

A: When dividing numbers with the same base, we subtract their exponents. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}.

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

A: When a number is raised to a negative exponent, we can rewrite it as a fraction with the base in the denominator and the exponent as a positive number. For example, a−m=1ama^{-m} = \frac{1}{a^m}.

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

A: To simplify an expression with multiple exponents, we need to apply the rules of exponents. For example, am⋅anap=am+n−p\frac{a^m \cdot a^n}{a^p} = a^{m+n-p}.

Q: What is the rule for raising a power to a power?

A: When raising a power to a power, we multiply their exponents. For example, (am)n=am⋅n(a^m)^n = a^{m \cdot n}.

Q: How do we simplify an expression with a fraction as an exponent?

A: To simplify an expression with a fraction as an exponent, we need to apply the rules of exponents. For example, amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the base is in the denominator and the exponent is positive.

Q: How do we simplify an expression with multiple fractions as exponents?

A: To simplify an expression with multiple fractions as exponents, we need to apply the rules of exponents. For example, amn⋅apq=am⋅q+n⋅pn⋅qa^{\frac{m}{n}} \cdot a^{\frac{p}{q}} = a^{\frac{m \cdot q + n \cdot p}{n \cdot q}}.

Q: What is the rule for simplifying an expression with a variable as an exponent?

A: When a variable is raised to a power, we can simplify the expression by applying the rules of exponents. For example, xm+n=xm⋅xnx^{m+n} = x^m \cdot x^n.

Q: How do we simplify an expression with a variable as a fraction as an exponent?

A: To simplify an expression with a variable as a fraction as an exponent, we need to apply the rules of exponents. For example, xmn=xmnx^{\frac{m}{n}} = \sqrt[n]{x^m}.

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

A: A rational exponent is a fraction as an exponent, while an irrational exponent is a non-integer exponent.

Q: How do we simplify an expression with a rational exponent?

A: To simplify an expression with a rational exponent, we need to apply the rules of exponents. For example, amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}.

Q: What is the rule for simplifying an expression with a radical as an exponent?

A: When a radical is raised to a power, we can simplify the expression by applying the rules of exponents. For example, amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}}.

Q: How do we simplify an expression with a radical as a fraction as an exponent?

A: To simplify an expression with a radical as a fraction as an exponent, we need to apply the rules of exponents. For example, ampn=amn⋅p\sqrt[n]{a^{\frac{m}{p}}} = a^{\frac{m}{n \cdot p}}.

Q: What is the difference between a real number and a complex number as an exponent?

A: A real number as an exponent is a number that can be expressed as a fraction or a decimal, while a complex number as an exponent is a number that can be expressed as a combination of real and imaginary parts.

Q: How do we simplify an expression with a complex number as an exponent?

A: To simplify an expression with a complex number as an exponent, we need to apply the rules of exponents and the properties of complex numbers. For example, am+ni=am⋅eni⋅ln⁡(a)a^{m+ni} = a^m \cdot e^{ni \cdot \ln(a)}.

Q: What is the rule for simplifying an expression with a trigonometric function as an exponent?

A: When a trigonometric function is raised to a power, we can simplify the expression by applying the rules of exponents and the properties of trigonometric functions. For example, sin⁡m(x)=sin⁡(m⋅arcsin⁡(x))\sin^m(x) = \sin(m \cdot \arcsin(x)).

Q: How do we simplify an expression with a trigonometric function as a fraction as an exponent?

A: To simplify an expression with a trigonometric function as a fraction as an exponent, we need to apply the rules of exponents and the properties of trigonometric functions. For example, sin⁡mn(x)=sin⁡(mn⋅arcsin⁡(x))\sin^{\frac{m}{n}}(x) = \sin(\frac{m}{n} \cdot \arcsin(x)).