Question

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

Short answer

The value of \(x\) is 27.

Step-by-step solution

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 \(3^{3} \times (3^2)^{12} = 3^{x}\).

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 \(3^{3} \times 3^{24} = 3^{x}\).

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 \(3^{27} = 3^{x}\).

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, \(x = 27\).

Key concepts

Understanding Exponents

Exponents are a shorthand way to express repeated multiplication of the same number. For example, \(3^3\) means you multiply 3 by itself two more times: \(3 \times 3 \times 3\). In general, \(a^n\) means you multiply \(a\) by itself \(n\) times. Exponents are powerful because they simplify expressions and make calculations easier to manage, especially as numbers get larger. It's important to remember some basic rules involve operations like multiplication, division, and taking powers of powers.

Power to Power Rule

When you raise a power to another power, you apply the power to power rule. This rule states that \((a^m)^n = a^{m \times n}\). The logic here is simple: each instance of \(a^m\) gets multiplied "n" times, so you just add up all those components, resulting in multiplying the exponents. For instance, if you have \((2^3)^2\), this means you have \(2^3\) multiplied by itself, thus becoming \(2^{3 \times 2} = 2^6\). This rule proves extremely useful in simplifying expressions with multiple layers of exponents.

Same Base Multiplication

In algebra, when two powers have the same base and are multiplied together, you can simplify them by adding their exponents. This is because multiplication of like bases accumulates their powers. Hence, for any same base \(a\), \(a^m \times a^n = a^{m+n}\). Let's break it down: if you have \(3^2 \times 3^4\), you're effectively combining the exponents, resulting in \(3^{2+4} = 3^6\). Recognizing this pattern simplifies problems and reduces the steps required to find solutions.

Working with Algebraic Equations

Algebraic equations allow us to solve for unknown variables using known quantities. In this context, equations involving exponents make setting the values equal an effective method of solving. For instance, if both sides of the equation have the same base number, the rule is that their exponents too must be equal in a valid equation. This means once we equate bases on both sides, the exponents become our equation to solve. Taking the exercise, \(3^{27} = 3^x\), we know immediately that \(x\) must be 27 since the bases are equivalent. Understanding this concept aids greatly in unraveling equations swiftly and accurately.