Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 4t^-2+8t^-4

Verified step by step guidance

1. Identify the terms in the expression: \(4t^{-2}\) and \(8t^{-4}\).
2. Determine the least power of the variable \(t\) in the expression, which is \(t^{-4}\).
3. Factor \(t^{-4}\) out of each term in the expression.
4. Rewrite each term by dividing by \(t^{-4}\): \(4t^{-2} = 4t^{-2} \div t^{-4}\) and \(8t^{-4} = 8t^{-4} \div t^{-4}\).
5. Express the factored form as \(t^{-4}(4t^{2} + 8)\).

Key Concepts

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In algebra, this often involves identifying common terms or expressions that can be extracted from a polynomial. For example, in the expression 4t^-2 + 8t^-4, we can factor out the least power of t, which is t^-4, to simplify the expression.

Least Power of a Variable

The least power of a variable in an expression refers to the term with the smallest exponent. When factoring, it is essential to identify this term to simplify the expression effectively. In the given example, the terms 4t^-2 and 8t^-4 have powers of -2 and -4, respectively, making t^-4 the least power, which is crucial for the factoring process.

Positive Real Numbers

In algebra, assuming that variables represent positive real numbers means that the values of these variables are greater than zero and can take any value on the real number line, excluding negatives and zero. This assumption is important when performing operations like factoring, as it ensures that the expressions remain valid and meaningful within the context of real number arithmetic.