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Chapter 11: Problem 3

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

Short Answer

Expert verified
In conclusion, the value of \(x\) is 27.

Step by step solution

01

Express 9 as a Power of 3

Since 9 is the square of 3, it can be expressed as \(3^2\). Therefore, we can rewrite \(9^{12}\) as \((3^2)^{12}\) in the given equation.
02

Use properties of Exponents

The law of exponentiation tells us that \((a^m)^n=a^{mn}\). We apply this rule to \((3^2)^{12}\) which is equal to \(3^{2 \times 12}\), and our equation becomes \(3^3 \times 3^{24} = 3^x\).
03

Combine like Bases

When multiplying exponents with the same base, we can simply add the exponent values together. This gives us \(3^{(3+24)}=3^x\). In other words, \(3^{27} = 3^x\).
04

Compare Exponent on Both Sides

Now that both sides of the equation have the same base, \(x\) must be equal to the power on the left hand side. Hence our solution is \(x=27\).

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Most popular questions from this chapter

The original selling price of an item at a store is 40 percent more than the cost of the item to the retailer. If the retailer reduces the price of the item by 15 percent of the original selling price, then the difference between the reduced price and the cost of the item to the retailer is what percent of the cost of the item to the retailer?

If \(A=2 x-(y-2 c)\) and \(B=(2 x-y)-2 c,\) then \(A-B=\) a. \(-2 y\) b. \(-4 c\) c. \(-0\) d. \(-2 y\) e. \(-4 c\)

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