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Chapter 10: Problem 62

Simplify. Write the answers with positive exponents only. $$\frac{m^{5}}{m^{7}}$$

Short Answer

Expert verified
\(\frac{1}{m^2}\)

Step by step solution

01

Subtract the Exponents

When dividing like bases, subtract the exponent in the denominator from the exponent in the numerator. So, for the expression \(\frac{m^5}{m^7}\), subtract 7 from 5.
02

Apply the Rule

Using the rule from Step 1, we get: \(m^{5-7} = m^{-2}\).
03

Convert Negative Exponent to Positive Exponent

To write the answer with positive exponents only, convert \(m^{-2}\) to \(\frac{1}{m^2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

positive exponents
In mathematics, **positive exponents** indicate how many times a number, called the base, is multiplied by itself. For example, in the expression \(a^3\), the base is \(a\), and the exponent is 3. This means \(a\) is multiplied by itself three times: \(a \times a \times a\).

  • The outcome is also known as the power of the base.
  • Positive exponents simplify calculations, especially with repeated multiplication.
Knowing how to handle positive exponents makes operations like multiplication and division of powers easier.
negative exponents
Negative exponents might seem tricky at first, but they follow straightforward rules. When you see a negative exponent, imagine it as an instruction to take the reciprocal of the base raised to the positive version of that exponent.

For instance, if you have \(a^{-3}\), it translates to \(\frac{1}{a^3}\).

  • Negative exponents indicate a division or a reciprocal action.
  • This concept helps in simplifying expressions and switching between forms easily.
Converting negative exponents to positive ones is essential in simplifying many algebraic expressions.
exponent rules
Understanding **exponent rules** is key to working with expressions involving powers. There are several core rules which you need to grasp:

  • Product of Powers Rule: When multiplying like bases, add their exponents. For example, \(a^m \times a^n = a^{m+n}\).
  • Power of a Power Rule: When raising a power to another power, multiply their exponents: \((a^m)^n = a^{m\cdot n}\).
  • Quotient of Powers Rule: When dividing like bases, subtract the exponent in the denominator from the exponent in the numerator: \(a^m / a^n = a^{m-n}\).
Each rule helps simplify operations and solve algebraic problems efficiently.
division of like bases
To simplify expressions with division of like bases, you use the **quotient of powers rule**. This rule states that when you divide two powers with the same base, you simply subtract the exponents.

For example, in \(\frac{m^5}{m^7}\):
  • First, write the operation with subtraction of exponents: \(m^{5-7}\).
  • Then, simplify to get \(m^{-2}\).
  • Finally, convert the negative exponent to a positive one by writing it as a fraction: \(\frac{1}{m^2}\).
This step-by-step method helps clarify the simplifying process, ensuring you handle exponents correctly.

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Most popular questions from this chapter

Subtract \(\left(9 x^{2}+16 x-4\right)\) from \(\left(2 x^{2}-4 x-9\right)\)

Simplify. Write the answers with positive exponents only. $$8^{2} 8^{-4}$$

Simplify the expression. $$\left(y^{2}\right)^{3}\left(y^{4}\right)^{2}$$

Simplify the expression. $$\left(2 p^{2}\right)^{3}(p q)^{2}$$

Write the opposite of the polynomial. $$2.4 m-5.1 n$$
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