Problem 14 - Algebra 1

Question

Evaluate the expression when \(x=3\) $$ (3 x)^{4} $$

Step-by-Step Solution

Answer

The evaluated expression is 6561.

Step 1: Substitute the variable

Replace the variable \( x \) in the expression with the given value which is 3. After substitution, the expression becomes \( (3*3)^{4} \).

Step 2: Simplify multiplication

Firstly, perform the multiplication inside the parentheses. \( 3*3 = 9 \). So, the expression is now \( 9^{4} \).

Step 3: Evaluate exponent

Finally, evaluate \( 9^{4} \). The result is 6561.

Key Concepts

Substitution in Evaluating Expressions

Substitution is like a game of matching where you replace a placeholder with its actual value. In the context of evaluating expressions, it involves replacing variables with given numbers to simplify the expression further. For example, if you are given an expression like (3x)^4 and told that x=3, the substitution means you replace all instances of x< / em > with 3. So, (3x)^4 becomes (3*3)^4 after substitution.

Why do we do this? Substitution provides a clear path to simplify the expression step by step. It also eliminates the variables so we can work with actual numbers, making calculations possible. When you substitute correctly, the rest of the problem often falls into place quite naturally.

Understanding Exponents

Exponents are like shortcuts in multiplication. Instead of saying you want to multiply a number by itself several times, you can use a small raised number (the exponent) to indicate how many times to multiply that number by itself. In the expression (3*3)^4, the exponent is 4, which tells us to multiply 9 (since 3 times 3 equals 9) by itself four times: \[9 \times 9 \times 9 \times 9\].

Exponents can be tricky because they pack a lot of punch—a small change in the exponent can make a big difference in the result. This is why understanding how they work is crucial. Always remember, an exponent applies only to the number it is directly connected to, unless parentheses indicate otherwise.

Simplify Multiplication by Recognizing Patterns

Simplifying multiplication makes large problems more manageable. It involves performing multiplication step by step and reducing expressions to their simplest form. For multiplication of the same numbers, as seen with exponents, recognizing patterns can speed up the process. After substituting x< / em> with 3 in the expression (3x)^4, you multiply 3 by itself to get 9. Then, you have 9^4, or 9 multiplied by itself four times.

The trick to simplify this multiplication is to do it in stages: 9 x 9 = 81, and then 81 x 81 instead of multiplying 9 by itself four times in one go. This helps you manage the large numbers one step at a time, reducing errors and making the process more understandable.