Question

Write the expression in exponential form. $$ 3 \cdot(t \cdot t \cdot t \cdot t) $$

Short answer

The expression in exponential form is \(3 \cdot t^4\).

Step-by-step solution

Identify the base and the exponent

Examine the expression. In \(3 \cdot(t \cdot t \cdot t \cdot t)\), the base is \(t\) and it is being multiplied by itself 4 times, which is the exponent.

Write the part with repeated multiplication in exponential form

The multiplication of \(t\) by itself four times can be rewritten as \(t^4\). So, the expression given simplifies to \(3 \cdot t^4\).

Finalize and check your answer

Check that your answer reflects the number of times \(t\) is being multiplied by itself, and it does. The final answer is \(3 \cdot t^4\).

Key concepts

Exponents

When you encounter repeated multiplication of the same number, you are dealing with exponents. An exponent, sometimes called a 'power', tells you how many times to multiply the base, which is the number being multiplied, by itself. For example, if you have the number 5 and you multiply it by itself 3 times, like this: \( 5 \times 5 \times 5 \), you can use exponential notation to simplify this expression. In this case, 5 is the base, and the number of times it's being multiplied, 3, is the exponent. This can be written compactly as \( 5^3 \).

When working with exponents, there are a few rules to keep in mind, such as the 'Product of Powers' rule which tells you that when you multiply two powers with the same base, you can add their exponents together. Simplification is key in using exponents effectively. It can make complex expressions much shorter and more manageable, allowing for easier computation and understanding.

Exponential Notation

Exponential notation is a way of writing numbers that are too large or too small to be conveniently written in decimal form. It is also used to simplify expressions involving repeated multiplication, as we've seen with exponents. In exponential notation, you write a number as a base raised to an exponent. The base is the number being repeatedly multiplied, and the exponent shows how many times the base is used as a factor.

For instance, in the expression \(3 \times(t \times t \times t \times t)\), exponential notation allows us to write it more succinctly as \(3 \times t^4\). The number 3 remains untouched because it is not part of the repeated multiplication. Recognizing how to write expressions in exponential form is essential for simplifying mathematical work and is commonly used in various areas of mathematics, including algebra and calculus.

Simplifying Expressions

Simplifying expressions means to alter them into their most concise and simplest form without changing their value. The process may involve combining like terms, using exponents, and applying other algebraic rules. The goal is to make the expression as clean and easy to work with as possible. Take the expression from our exercise \(3 \times(t \times t \times t \times t)\). Simplifying this expression involves seeing that the letter \(t\) is multiplied by itself four times, and thus, can be written using an exponent. The simplified form, \(3 \times t^4\), is much easier to read and work with for subsequent operations.

Simplification is not just about making expressions shorter; it's about clarity and efficiency in mathematics. A simplified expression minimizes the potential for errors in complex calculations and makes it easier for others to understand the work that has been done. Therefore, mastering the art of simplifying expressions is a fundamental skill for any student of mathematics.