Problem 62 - Beginning and Intermediate Algebra

Question

Simplify. Assume that the variables represent nonzero integers. $$\frac{m^{10 u}}{m^{3} u}$$

Step-by-Step Solution

Answer

The simplified expression for \(\frac{m^{10 u}}{m^{3} u}\) is \(m^{7u}\).

Step 1: Rewrite the expression using the property of exponents

We can rewrite the expression using the property of exponents mentioned above: \( \frac{m^{10 u}}{m^{3} u} = \frac{m^{10 u}}{1} \cdot \frac{1}{m^{3} u} = m^{10 u} \cdot m^{-3 u}\)

Step 2: Simplify the expression by combining exponents

Having written the expression in this form, we can now simply combine the exponents, using the property \(a^{m} \cdot a^{n}= a^{m+n}\): \[m^{10 u} \cdot m^{-3 u} = m^{10u - 3u}\]

Step 3: Calculate the new exponent

Now, we can subtract the exponents to complete the simplification: \[m^{10u - 3u} = m^{7u}\] So the simplified expression is \(m^{7u}\).

Key Concepts

Simplifying Expressions

When working with algebraic expressions, particularly those that involve exponents, it is essential to simplify them to make calculations easier and more readable. Simplifying expressions means reducing them to their most basic form without changing their value. In the original exercise, the expression given is \(\frac{m^{10u}}{m^{3}u}\). This expression can be made simpler by applying exponent rules.

• The division of expressions with the same base can be simplified by subtracting the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• This means dividing two terms with the same base results in subtracting the powers involved.

Applying these rules enabled us to transform the complex fraction into a single exponential expression, \(m^{7u}\), which is much more straightforward to work with. Simplification using these methods is important in algebra as it makes expressions easier to evaluate or compare.

Properties of Exponents

Understanding properties of exponents is crucial for simplifying complex algebraic expressions. These properties include rules that dictate how to handle expressions with the same base when multiplying or dividing. Let’s look at some key properties:

• Product of Powers: When you multiply terms with the same base, add their exponents: \(a^m \cdot a^n = a^{m+n}\).
• Quotient of Powers: When you divide terms with the same base, subtract the exponent of the denominator from that of the numerator: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power: When raising a power to another exponent, multiply the exponents: \((a^m)^n = a^{m \cdot n}\).

In our example, we applied the quotient of powers property to simplify the given expression. This property is an efficient way to handle problems involving repeated multiplication or division of the same base.

Integer Exponents

Integer exponents are a fundamental part of algebra, frequently encountered in mathematical expressions. They help describe how many times to multiply a number by itself. An integer exponent can be positive, zero, or negative:

• Positive Exponents: Indicate how many times the base is multiplied by itself, such as \(a^3 = a \cdot a \cdot a\).
• Zero Exponents: Any non-zero base raised to the zero power is 1, so \(a^0 = 1\).
• Negative Exponents: Represent reciprocal powers, meaning \(a^{-n} = \frac{1}{a^n}\).

In the exercise provided, we noticeably used a negative exponent \(m^{-3u}\) to move terms from the denominator to the numerator for simpler calculation. This conversion is commonplace when managing fractions or expressions with exponents across different positions. Understanding integer exponents is crucial since it is applied widely in algebra, calculus, and other mathematical calculations.