Question

Write each expression without parentheses. Assume all variables are positive. $$ \left(4 b^{2}\right)^{2 t} $$

Short answer

Question: Rewrite the expression \((4b^2)^{2t}\) without parentheses. Answer: \(4^{2t}b^{4t}\)

Step-by-step solution

Apply the power of a power rule

The power of a power rule states that \((a^m)^n = a^{m \cdot n}\). So, in our expression, we have \((4b^2)^{2t}\). We can rewrite this as \(4^{2t} \cdot (b^2)^{2t}\).

Simplify the expression further

Now, we will apply the power of a power rule again for \((b^2)^{2t}\), which gives us \(b^{2 \cdot 2t}\). Now our expression is \(4^{2t} \cdot b^{4t}\).

Rewrite the expression without parentheses

Finally, we have obtained the expression without any parentheses: $$4^{2t}b^{4t}$$.

Key concepts

Exponents

Exponents are a fundamental part of algebra and mathematics. They are used to represent repeated multiplication of a base number. You might encounter them often when dealing with expressions like \(a^n\), where \(a\) is the base and \(n\) is the exponent. The expression represents \(a\) multiplied by itself \(n\) times.

In the expression \( (4b^2)^{2t} \), we are dealing with an exponent applied to a power, which involves using the power of a power rule. This rule simplifies expressions by multiplying exponents: \((a^m)^n = a^{m \cdot n}\).

Using the rule helps us transform and simplify complex expressions, making calculations more manageable. In our example, \((b^2)^{2t}\) simplifies to \(b^{2 \cdot 2t}\), which makes the usage of exponents clear and straightforward.

Simplifying Expressions

Simplifying expressions means making them as "simple" or compact as possible without altering their value. This often involves reducing expressions by applying mathematical rules and operations.

When simplifying exponential expressions, you use different properties like the power of a power rule, product rule, and quotient rule. For the expression \((4b^2)^{2t}\), simplifying involves removing parentheses by applying these rules.

- Apply power of a power rule to both terms: \((4)^{2t} \cdot (b^2)^{2t} \).- Simplify each component: \(4^{2t}\) and \(b^{4t}\).

The goal is to express everything in terms of a single base raised to a power, making further mathematical operations more efficient and less complex.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. These expressions can range from simple, like \(x + 2\), to more complex ones involving exponents and multiple terms.

The task typically involves variables which represent numbers, and these can be manipulated like ordinary numbers. In the expression \( (4b^2)^{2t} \), \(b\) is a variable and \(t\) is another that might represent a constant number or a further exponent.

Working with algebraic expressions often involves converting them into an easier format for understanding or calculation. By changing \((4b^2)^{2t}\) into \(4^{2t}b^{4t}\), we have removed the parentheses and made the expression easier to manage and evaluate. Thus, understanding and manipulating algebraic expressions is crucial for solving real-world math problems efficiently.