Question

Write each expression as a power raised to a power. There may be more than one correct answer. $$ (x+3)^{2 w} $$

Short answer

Answer: The expression can be written as \((x+3)^{w \cdot 2}\) to represent it as a power raised to another power.

Step-by-step solution

Rewrite the exponent In the given expression \((x+3)^{2w}\), we are going to rewrite the exponent \(2w\):

Note that \(2w\) can also be written as \(w \cdot 2\). Now we can use the new exponent and rewrite the expression.

Express the power raised to a power

Substitute the rewritten exponent in the expression: $$ (x+3)^{w \cdot 2} $$ So the expression can be written as a power raised to another power if we rewrite the exponent as the product of two numbers which are themselves exponents. In this case, \((x+3)^{w \cdot 2}\) is another way to express the original expression \((x+3)^{2w}\).

Key concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases comprised of numbers, variables, and operations. They are the building blocks of more complex equations and allow us to represent mathematical situations in a concise way.

For example, in the expression \((x+3)^{2w}\), we see a combination of a variable \(x\), constants \(3\), and an exponent \(2w\). This expression tells us to take \(x + 3\) as a base and raise it to the power of \(2w\). Such expressions enable us to generalize patterns and find solutions to problems involving unknown quantities.

• Variables: Symbols like \(x\) that represent numbers whose exact values are not yet known.
• Constants: Fixed numbers like \(3\) that do not change.
• Operations: Include addition, subtraction, multiplication, and exponentiation.

Understanding algebraic expressions allows you to manipulate and simplify them to solve equations effectively.

Power Rules

Power rules are essential in algebra, especially when dealing with exponents. These rules help us simplify expressions and solve equations involving powers more efficiently. In this particular exercise, the rule we use is about a power raised to a power.

The expression \((x+3)^{2w}\) demonstrates the rule where an exponent is itself an exponent, allowing transformation into a format \((a^{m})^{n} = a^{m \cdot n}\).

• Product of Powers: When multiplying like bases,
  \(a^m \times a^n = a^{m+n}\).
• Power of a Power: When raising a power, multiply the exponents,
  \( (a^m)^n = a^{m \cdot n} \).
• Power of a Product: Each factor of the product is raised to the power,
  \( (ab)^m = a^m b^m \).

By applying these rules, we can rewrite expressions like \((x+3)^{2w}\) as \((x+3)^{w \cdot 2}\), thus showing the flexibility of expressions.

Mathematical Notation

Mathematical notation is the language of mathematics, using symbols to represent numbers, operations, and relationships. It provides a universal way of communicating complex mathematical ideas in a concise and unambiguous manner.

In the expression \((x+3)^{2w}\), notation helps convey complex ideas succinctly, allowing mathematicians to understand and manipulate expressions with ease.

• The exponentiation symbol \(^{\ }\), which tells us to raise a number to a specified power.
• Parentheses \(()\) indicate the terms to be considered as a single unit when performing operations.
• Variables like \(x\) represent unknown values that can change depending on the problem.

Mastery of mathematical notation is crucial. It enables students to interpret and solve problems systematically, thus showcasing the power of clear and precise mathematical communication.