Question

Evaluate the expression for the given value of the variable. $$ 6 t^{4} \text { when } t=3 $$

Short answer

The value of the expression \(6t^{4}\) when \(t = 3\) is 486.

Step-by-step solution

Identify the Given Information

First, identify what's given in the problem. The expression is \(6t^{4}\) and the value of the variable \(t\) is 3.

Substitute the Value of the Variable

Next, replace every instance of \(t\) with 3 in the expression. So, the expression will be \(6 \cdot 3^{4}\).

Simplify the Expression

Now simplify the expression. First, calculate \(3^{4}\), which equals 81. Then multiply 81 by 6, which equals 486.

Key concepts

Substitution

When evaluating expressions, substitution is your first step; think of it as the foundation that sets the stage for simplification and solving. It involves replacing variables in an expression, like 't', with their given numerical values. In the example where we have the expression
\( 6t^{4} \) with 't' equal to 3, we substitute 't' with 3 to get
\( 6 \times 3^{4} \). This basic operation is essential, as it turns abstract formulas into concrete numbers we can work with. If you find substitution tricky, remember, it's a direct swap: every 't' becomes a '3' in this case. Try to write down the expression with the substitutions before proceeding to any calculations, as this will help prevent mistakes and make the process clearer.

Exponential Expressions

Understanding exponential expressions is key in various areas of mathematics, including our example. An exponent, such as the '4' in \(3^{4}\), tells us how many times to multiply the base, '3', by itself. It's not \(3 \times 4\), but \(3 \times 3 \times 3 \times 3\). When simplifying, you always want to handle the exponential part first before any multiplication that is outside, as was done in the step where \(3^{4}\) was calculated to be 81 before it got multiplied by 6. There's often confusion around this, but if you remember to work on the exponent first, you’ll stay on the right track. For those still learning, using a calculator for the exponential part is perfectly okay, but practicing with smaller numbers can help build your confidence.

Simplifying Expressions

Once substitution is done, simplifying the expression comes next. Simplification involves executing the math operations in the correct order, often guided by the BODMAS/BIDMAS rule (Brackets, Orders, Division and Multiplication, Addition and Subtraction). In our expression \(6 \times 3^{4}\), after calculating the exponent, we proceed by multiplying the result by 6. Keeping these steps separate and clear ensures accuracy. Additionally, simplifying expressions isn't just about getting to an answer; it's about understanding why that answer is what it is. It's about the journey, not just the destination. If you're finding the simplification process confusing, always break down the steps and tackle them one at a time. It’s also helpful to write down each phase of your calculation, so you can follow your own thought process and spot where you might have gone wrong if the need arises.