Mastery Check Solutions

Show What You Know

Both groups of students incorrectly solved the problem: $64 \cdot 4^{2 (x + 2)} = 2^{8 x + 6}$

Read the entire problem before starting. You may prefer to solve the equation and then identify the errors, or you may prefer to find the errors and then solve it correctly.

Solve the problem to find the correct solution. (Hint: Review the exponent rules on your Formula Sheet.)

$\begin{array}{rcl} 2^{6} \cdot 2^{2 (2 (x + 2))} & = & 2^{8 x + 6} \\ 2^{6} \cdot 2^{4 (x + 2)} & = & 2^{8 x + 6} \\ 6 + 4 (x + 2) & = & 8 x + 6 \\ 6 + 4 x + 8 & = & 8 x + 6 \\ 4 x + 14 & = & 8 x + 6 \\ 8 & = & 4 x \\ x & = & 2 \end{array}$

Note

Q: How do you correctly combine exponents when the common bases (on the left side) are multiplied?

A: Add the exponents together

Remember to check your work (by hand or with technology) to prove your answer is correct.

Explain the error each group made.

Sample: The Rationals group multiplied the exponents on the left side of the equation instead of adding them. The Irrationals group wrote $4 = 2$ instead of $4 = 2^{2}$ .

Change the original equation to an inequality so that the incorrect answers for both the Rationals Group and Irrationals Group are true.

$64 \cdot 4^{2 (x + 2)} > 2^{8 x + 6} or 64 \cdot 4^{2 (x + 2)} \geq 2^{8 x + 6}$

$\begin{array}{rcl} 2^{6} \cdot 2^{4 (x + 2)} & > & 2^{8 x + 6} \\ 6 + 4 (x + 2) & > & 8 x + 6 \\ 6 + 4 x + 8 & > & 8 x + 6 \\ 4 x + 14 & > & 8 x + 6 \\ 8 & > & 4 x \\ x & < & 2 \end{array}$

Note

Both groups found incorrect solutions that were less than 2; therefore $x < 2$ . So the Rationals and Irrationals groups have true answers when the equation is rewritten as an inequality. Your student may need to write out the work to prove the direction of the inequality.

Say What You Know

In your own words, talk about what you have learned using the objectives for this lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

Solve exponential equations.
Solve exponential inequalities.