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Chapter 11: Problem 3

If \(3^{3} \times 9^{12}=3^{x},\) what is the value of \(x ?\)

Short Answer

Expert verified
The value of \(x\) is 27.

Step by step solution

01

Writing 9 as a Power of 3

First, write 9 as a power of 3. Note that \(9 = 3^2\). So, the exercise can be rewritten as \(3^{3} \times (3^2)^{12} = 3^{x}\).
02

Applying the Power to Power Rule

Next, use the power to power rule, where \((a^m)^n = a^{mn}\). This turns \((3^2)^{12}\) into \(3^{2 \times 12}\) or \(3^{24}\). So, the equation becomes \(3^{3} \times 3^{24} = 3^{x}\).
03

Power Rule for Same Bases

Then, apply the rule that states when multiplying two expressions with the same base, the exponents are added. This turns \(3^{3} \times 3^{24}\) into \(3^{3 + 24}\) or \(3^{27}\). Thus, the equation is \(3^{27} = 3^{x}\).
04

Setting Exponents Equal to Each Other

Finally, since the bases (3) of each side of the equation are equal, we can say that the exponents must also be equal. Therefore, \(x = 27\).

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Most popular questions from this chapter

If \(x=3 a\) and \(y=9 b,\) then all of the following are equal to \(2(x+y)\) EXCEPT a. \(3(2 a+6 b)\) b. \(6(a+3 b)\) c. \(24\left(\frac{1}{4} a+\frac{3}{4} b\right)\) d. \(\frac{1}{3}(18 a+54 b)\) e. \(12\left(\frac{1}{2} a+\frac{3}{4} b\right)\)

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