Question

Remove the brackets from the expression \(\left(4 x^{3}\right)^{5}\)

Short answer

Question: Remove the brackets from the given expression: \(\left(4x^3\right)^5\). Answer: The expression without brackets is \(1024x^{15}\).

Step-by-step solution

Identify the power of a power rule

We will use the power of a power rule \((a^m)^n = a^{mn}\) on the expression \(\left(4x^3\right)^5\).

Apply the power rule to the expression

Applying the rule to the given expression, we have: \(\left(4x^3\right)^5 = 4^5x^{3\cdot5}\).

Solve for the expressions

Now, we can simplify the powers: \(4^5 = 1024\) and \(3 \cdot 5 = 15\)

Write the simplified expression

Putting the simplified powers back into the expression, we get: \(1024x^{15}\). So, the expression without brackets is \(1024x^{15}\).

Key concepts

Simplifying Expressions

In algebra, one of the key tasks is to simplify expressions to make them easier to understand and work with. Simplifying an expression involves reducing it to its least complex form while maintaining its original value. This process often includes applying various algebraic rules and properties, such as the distributive property, combining like terms, and in the case of our exercise - the power of a power rule.

• Start by recognizing the structure of the expression and identify applicable algebraic rules.
• Apply these rules methodically, step by step, to avoid mistakes and ensure every part of the expression is simplified appropriately.
• Check your work by substituting numbers into the original and simplified expressions to see if they yield the same value.

When approached systematically, simplifying expressions is a skill that becomes increasingly intuitive with practice. This foundational skill sets the stage for solving more complex equations and understanding advanced concepts in algebra.

Exponentiation

Exponentiation is an arithmetic operation that involves raising a number, known as the base, to the power of an exponent. It's a shorthand for repeated multiplication and plays a crucial role in algebra. The power of a power rule is a specific instance of exponentiation where an exponent is raised to another exponent.

To use this rule, simply multiply the exponents together. For example, the expression \(\left(a^m\right)^n\) becomes \(a^{mn}\) when applying the power of a power rule. Here's a simple bullet-list to recap:

• The base \(a\) remains the same.
• The exponents \(m\) and \(n\) are multiplied to form the new exponent \(mn\).
• This rule is essential for simplifying expressions with multiple layers of powers.

Understanding exponentiation and rules like the power of a power is fundamental in algebra and is frequently applied in various fields of study, including engineering and sciences.

Algebra for Engineers

Algebra is an indispensable tool for engineers, providing the means to model, analyze, and solve real-world problems. It facilitates the description of relationships between variables and the manipulation of mathematical models essential to engineering design and analysis.

Application in Engineering

In the context of our exercise, an engineer might use the power of a power rule to simplify the calculation of material properties, electronic circuit analyses, or in the sizing of components in structural engineering.

Engineering often requires working with formulas that involve powers and exponents. Whether it's calculating loads, resistances, or energy consumption, an engineer must be adept at transforming complex expressions into simpler, more digestible forms.

• Develop a strong foundation in algebraic principles.
• Practice applying these principles to various engineering problems.
• Ensure accuracy in calculations by methodically simplifying expressions.

Algebra serves as a language that engineers speak fluently to develop innovative solutions and optimize systems in every aspect of modern engineering.