Problem 76 - Algebra 1

Question

Evaluate the expression. $$\left(3^{3}\right)^{2}$$

Step-by-Step Solution

Answer

The evaluated expression \((3^{3})^{2}\) equals to 729.

Step 1: Apply the Power of a Power Rule

According to the rule of exponents, when raising a power to a power, you multiply the exponents. So, for \((3^{3})^{2}\), you would multiply the exponent 3 by 2.

Step 2: Solve the multiplication

Multiply the exponents from the previous step. So we have \(3^{3*2}\), which equals to \(3^{6}\). This is a simplification of the expression.

Step 3: Evaluate the expression

Now, evaluate \(3^{6}\). To do this, multiply the base, 3, by itself 6 times, i.e., \(3*3*3*3*3*3\), which results in 729.

Key Concepts

Power of a Power Rule

Understanding the power of a power rule is like having a secret key to unlock complex exponentiation problems. Picture this: When we want to raise an exponent to another power, we can simplify our work by using this magical rule. Instead of doing long and tedious multiplications, we simply multiply the exponents together.

Take, for example, the expression \( (3^3)^2 \). It looks a bit daunting, right? But with the power of a power rule, we see it in a new light. The rule tells us to multiply the outer exponent, 2, with the inner exponent, 3, which gives us 2*3, resulting in a single exponent of 6. Suddenly, our expression becomes a much friendlier \(3^6\).

Exponents

The concept of exponents is all about repeated multiplication. It's like having an army of clones—when you have an exponent, it tells you how many times the base number should be multiplied by itself. The number before the exponent is the base, and the exponent itself, also known as the power, is how many times we use the base in multiplication.

When we break down an expression like \(3^6\), what we're really looking at is 3 multiplied by itself 6 whole times—\(3 \times 3 \times 3 \times 3 \times 3 \times 3\). It's a shortcut so we don't have to write long multiplication strings, saving time and paper. Knowing how exponents work can simplify many mathematical tasks, from solving algebraic expressions to understanding scientific notations.

Evaluate Expressions

Evaluating expressions is like being a math detective, where you investigate and simplify until you find the most simplified form. With our expression \(3^6\), our mission is to discover its numerical value. By evaluating, we're systematically carrying out the exponents literally— so we line up our 3s side by side and multiply them just as we laid out: 3 times 3, then that result times 3, and so on, until we've done it 6 times.

Our hard work pays off, and we reveal that \(3^6 = 729\), the final answer. This process is crucial not only in basic mathematics but also in various applications like calculating compound interest, understanding the growth of populations, or even in the realms of physics with exponential decay. Evaluating expressions is a key skill that helps turn mathematical scribbles into meaningful numbers.