Problem 6 - Algebra 1

Question

Use the quotient of powers property to simplify the expression. $$ \frac{m^{5}}{m^{11}} $$

Step-by-Step Solution

Answer

The simplified expression is \(1/m^{6}\).

Step 1: Identify the Base and Exponents

The base in both the numerator and the denominator is 'm'. The exponent in the numerator is 5 and the exponent in the denominator is 11.

Step 2: Apply the Quotient of Powers Property

The quotient of powers property states that \(a^{m}/a^{n} = a^{m-n}\). Thus, we subtract the exponent in the denominator from the exponent in the numerator: \(5 - 11 = -6\). The expression can now be rewritten as \(m^{-6}\).

Step 3: Simplifying Negative Exponent

Negative exponents can be simplified by placing the base with its exponent in the denominator of a fraction. This will create a positive exponent. So, the final simplified expression is \(1/m^{6}\).

Key Concepts

Negative Exponents

When dealing with exponents, a negative exponent signals the reciprocal of the base raised to the corresponding positive exponent.
For instance, when we see an expression like \(m^{-6}\), it implies that instead of multiplying \(m\) by itself 6 times, we are actually dividing by \(m\) 6 times.
In simpler terms, \(m^{-6}\) is equivalent to \(\frac{1}{m^{6}}\).

• Negative exponents do not mean negative numbers; they just represent the reciprocal.
• The positive exponent tells us how many times to divide.

Transferring the number to the denominator and changing the exponent to positive is key to simplifying an expression with a negative exponent. This concept is crucial when simplifying expressions in algebra.

Simplifying Expressions

Simplifying mathematical expressions involves reducing them to their simplest form without changing their value. This often means combining like terms, using properties of operations like the quotient of powers property, and simplifying exponents whenever possible.

• An expression is considered simplified if it is easier to read and interpret.
• Efficiency in mathematical communication can often boil down to how well expressions are simplified.
• In certain cases, simplification means reducing fractions to their lowest terms.

For example, starting from \(\frac{m^{5}}{m^{11}}\) and using exponent rules, we derived \(m^{-6}\), then further simplified it to \(\frac{1}{m^{6}}\). Each step makes the expression easier to understand and evaluates more straightforwardly.

Exponent Rules

Exponent rules are a set of guidelines that help us manage powers of numbers efficiently.
The rules simplify complex expressions that contain exponents, and they are especially handy when performing operations like multiplication and division on exponential terms.

• The product of powers rule: When multiplying same bases raised to exponents, add the exponents \(a^{m} \times a^{n} = a^{m+n}\).
• The power of a power rule: When taking an exponent to another power, multiply the exponents \((a^{m})^{n} = a^{m\cdot n}\).
• The quotient of powers rule: When dividing same bases raised to exponents, subtract the exponents \(a^{m}/a^{n} = a^{m-n}\).

In our original problem, \(\frac{m^{5}}{m^{11}}\), the quotient of powers rule led us to subtract the exponents, resulting in \(m^{-6}\). Understanding these rules makes simplifying expressions faster and reduces the possibility of errors.