Problem 71 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ \frac{10 m^{\Delta}}{5 m^{\text {t }}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression: $\frac{10m^{\Delta}}{5m^{\text {t }}}$ Answer: $2m^{\Delta-\text{t}}$

Step 1: Simplify the fraction

First, we need to divide the constants in the fraction: $$ \frac{10}{5} = 2 $$ The simplified problem now becomes: $$ 2 \times \frac{m^{\Delta}}{m^{\text {t }}} $$

Step 2: Apply power rules

Now, we have to simplify the variable part. We will use the quotient rule: $$ \frac{m^{\Delta}}{m^{\text {t }}} = m^{\Delta - \text {t }} $$

Step 3: Combine the results

Finally, combine the simplified fraction and variable part for the final solution: $$ 2 \times m^{\Delta - \text {t }} $$ So the simplified expression is: $$ 2m^{\Delta-\text{t}} $$

Key Concepts

Power Rules

Understanding power rules for exponents is crucial in simplifying algebraic expressions. When we talk about power rules, we refer to shortcuts that help us manage expressions with exponents efficiently. The key rules to remember are:

• Product of Powers: When multiplying two expressions with the same base, add the exponents: \( a^m \times a^n = a^{m+n} \)
• Power of a Power: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{mn} \)
• Power of a Product: To raise a product to a power, distribute the power to each factor: \( (ab)^n = a^n b^n \)

In the exercise, we didn't need these rules explicitly, but they form the foundation for understanding how to manipulate expressions that involve exponents. By mastering these, you can simplify complex problems more easily.

Quotient Rule

The quotient rule is extremely useful when you need to divide terms with exponents that have the same base. This rule states: when dividing, simply subtract the exponent of the denominator from the exponent of the numerator: \( \frac{a^m}{a^n} = a^{m-n} \).
In our original exercise, we applied this rule to simplify \( \frac{m^{\Delta}}{m^{\text{t}}} \), resulting in \( m^{\Delta - \text{t}} \). This subtraction of exponents is only valid when the bases are identical, which is an essential condition.
Here are some examples to illustrate:

• \( \frac{x^7}{x^3} = x^{7-3} = x^4 \)
• \( \frac{y^{10}}{y^{5}} = y^{10-5} = y^5 \)

Always make sure the bases are non-zero to avoid undefined expressions.

Algebra Simplification

Algebra simplification involves reducing expressions to their simplest form, eliminating unnecessary complexity. It requires combining coefficients, like terms, and using the rules of exponents when applicable.
In our task, we started by simplifying the fraction of constants: \( \frac{10}{5} = 2 \). This step simplifies numerical parts before moving onto variables, making subsequent calculations easier.
Having simplified the constants, the next step was to apply the quotient rule to the variables, giving us \( m^{\Delta - \text{t}} \). This results in the simplest form of the expression: \( 2m^{\Delta-\text{t}} \).
Remember:

• Combine the coefficients first.
• Use exponent rules to simplify variable parts.
• Always check your results to ensure accuracy.

Mastering these strategies helps in not only solving exercises but also in understanding broader algebraic concepts more clearly.