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🔟elementary algebra review

key term - $(a \times b)^n$

Citation:

Definition

$(a \times b)^n$ represents the product of two numbers, $a$ and $b$, raised to the power of $n$. This expression utilizes the multiplication properties of exponents, specifically demonstrating how to handle the exponent when multiplying two base values together. Understanding this term is crucial as it sets the foundation for manipulating expressions involving powers and applying properties that simplify calculations in algebraic contexts.

5 Must Know Facts For Your Next Test

  1. $(a \times b)^n$ can be rewritten using the Power of a Product Property as $a^n \times b^n$, allowing for simpler calculations.
  2. When evaluating $(a \times b)^0$, the result is always 1, as long as $a$ and $b$ are not zero, due to the rules of exponents.
  3. The expression $(a \times b)^{-n}$ can be simplified to $ rac{1}{(a \times b)^n}$, indicating that negative exponents represent reciprocal values.
  4. When multiplying several terms with exponents, such as $(x \times y \times z)^n$, it can be expanded to $x^n \times y^n \times z^n$ using the same property.
  5. In practical applications, $(a \times b)^n$ is often used in geometry and physics to calculate volumes and other properties involving products of lengths or forces.

Review Questions

  • How can you apply the Power of a Product Property to simplify the expression $(2 \times 3)^4$?
    • To simplify $(2 \times 3)^4$, you can use the Power of a Product Property, which states that $(xy)^n = x^n \times y^n$. Thus, $(2 \times 3)^4$ becomes $2^4 \times 3^4$. Calculating those values gives you $16 \times 81$, which equals 1296. This demonstrates how breaking down complex expressions can make calculations easier.
  • If $(x \times 5)^3 = 125x^3$, what does this tell you about the relationship between x and the expression?
    • $(x \times 5)^3 = 125x^3$ shows that when expanded using the Power of a Product Property, we get $x^3 \times 5^3$. Since $5^3 = 125$, it confirms that multiplying the base $x$ by 5 and raising it to the third power results in a term directly proportional to $x^3$. This connection illustrates how altering either factor impacts the overall value of the expression.
  • Discuss how understanding $(a \times b)^n$ can help solve real-world problems involving volume calculations.
    • Understanding $(a \times b)^n$ is crucial when solving real-world problems such as calculating the volume of a rectangular prism. For example, if each side of a cube has a length represented by $s$, then its volume can be expressed as $(s \times s \times s) = s^3$. Using the properties of exponents, knowing that $(a \times b)^3 = a^3 \times b^3$ helps when combining measurements from different dimensions. This knowledge enables accurate calculations in various fields like architecture and manufacturing, where precise volume measurements are essential.