Work out the values of t and w in the equalities below. a) 18^t / 18^15 = 18^5 b) 19^14 / 19^w = 19^2
Understand the Problem
The question asks to calculate the values of the variables t and w in the provided exponential equations. The first equation involves a fraction with base 18, while the second equation involves a fraction with base 19. We will solve each equation separately.
Answer
$t = 20$, $w = 12$
Answer for screen readers
The values are $t = 20$ and $w = 12$.
Steps to Solve
- Simplifying the first equation
For the first equation, use the property of exponents: $$ \frac{a^m}{a^n} = a^{m-n} $$ Thus, we rewrite the equation: $$ \frac{18^t}{18^{15}} = 18^{t-15} $$ So, the equation becomes: $$ 18^{t-15} = 18^{5} $$
- Setting the exponents equal
Since the bases are equal, set the exponents equal to each other: $$ t - 15 = 5 $$
- Solving for t
Now, solve for $t$: $$ t = 5 + 15 = 20 $$
- Simplifying the second equation
For the second equation, apply the same property of exponents: $$ \frac{19^{14}}{19^{w}} = 19^{14 - w} $$ Thus, the equation becomes: $$ 19^{14 - w} = 19^{2} $$
- Setting the exponents equal
Set the exponents equal to each other again: $$ 14 - w = 2 $$
- Solving for w
Now, solve for $w$: $$ -w = 2 - 14 $$ $$ -w = -12 \implies w = 12 $$
The values are $t = 20$ and $w = 12$.
More Information
This problem demonstrates the use of properties of exponents, specifically how to manipulate and equate powers with the same base. Understanding these properties helps simplify complex equations.
Tips
- Forgetting to apply the exponent rules correctly when simplifying fractions.
- Neglecting to set the exponents equal when the bases are the same.
- Overlooking algebraic simplifications when solving for variables.
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