Question

Then evaluate the expression when \(a=1\) and \(b=2\). $$ \left(a^{3}\right)^{2} $$

Short answer

The value of the expression \( (a^{3})^{2} \) when \(a=1\) is 1.

Step-by-step solution

Substitute the given values into the expression

In the given expression \( (a^{3})^{2} \), replace \(a\) with 1.

Apply the power of a power rule

In the expression \( (1^{3})^{2} \), the power of a number is found by multiplying the exponents. Thus, the exponent becomes \( 1^{3*2} \) or \( 1^{6} \).

Simplify the expression

Since any non-zero number raised to power 0 or higher is always 1, \( 1^{6} \) simplifies to 1. Note that \(1^6\) simply represents 1 multiplied by itself 6 times.

Key concepts

Exponent Multiplication

Exponent multiplication is a fundamental concept in algebra that deals with powers of numbers and variables. When an exponential expression is raised to another power, such as \( (a^{b})^{c} \), the powers are multiplied together. In the case of our example, \( (a^{3})^{2} \), we have a base of \( a \) raised to the 3rd power, which is then raised again to the 2nd power. According to the power of a power rule, you multiply the exponents: \( 3 \times 2 \), resulting in \( a^{6} \).

Remember, this rule holds true for any real numbers, not just integers, and it also applies when variables are used instead of specific numbers. It simplifies the process of dealing with multiple layers of exponents, making it a powerful tool for managing complex algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that represent a particular mathematical relationship or value. They can range from simple expressions with a single variable, like \( 2x \), to more complex ones involving multiple terms and exponents, such as \( 4x^{2} + 3xy - 5 \).

When dealing with expressions that include exponents, understanding the foundational rules like the power of a power rule simplifies the process of manipulation. In algebra, it's essential to use these rules to rewrite expressions in their simplest form. By doing so, one can solve equations, evaluate expressions for given variable values, and analyze functions more effectively. The beauty of algebraic expressions lies in their ability to generalize mathematical truths, regardless of the specific numbers involved.

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra which involves reducing an expression to its most basic form without changing its value. To simplify an expression, one needs to combine like terms, use distributive properties, and apply relevant exponent rules such as the power of a power rule and others like the product of powers rule and the zero exponent rule.

Simplifying might involve several steps, such as expanding expressions using the distributive property, factoring, and finally, reducing the expression by canceling out terms if possible. In the context of our exercise, the simplification step involved recognizing that any number, except zero, raised to any power is itself, making \( 1^{6} \) collapse to 1. Simplification enables us to present expressions in the most compact and understandable form which is particularly useful when dealing with complex or higher-level mathematics.