Verified Answer :

  • For \((3x^2y)(5x)\), multiply coefficients and add exponents for \(x\), resulting in \(\boxed{15x^3y}\).
  • For \((x^2y^3)(x^{-2}y^{-2})\), combine bases with the product rule, giving \(x^0y^1 = \boxed{y}\).
  • For \(\frac{x^3}{x^{-2}}\), use the quotient rule \(x^{m-n}\) to get \(x^{3-(-2)} = \boxed{x^5}\).
  • For \((2x^{-1})^3\), distribute the power and convert negative exponents to positive to get \(\boxed{\frac{8}{x^3}}\).

Explanation

  1. Understanding the Properties of Exponents

The goal is to simplify four algebraic expressions involving exponents. The properties of exponents we will utilize include:

  • The product rule: \(a^m \cdot a^n = a^{m+n}\).
  • The quotient rule: \(\frac{a^m}{a^n} = a^{m-n}\).
  • The power of a power rule: \((a^m)^n = a^{m \cdot n}\).
  • The power of a product rule: \((ab)^n = a^n \cdot b^n\).
  • The negative exponent rule: \(a^{-n} = \frac{1}{a^n}\) for \(a eq 0\).

Each final expression must only contain positive exponents.

  1. Simplifying Expression (a): \((3x^2y)(5x)\)

To simplify \((3x^2y)(5x)\), we group the coefficients and the like variables together:
\[(3 \times 5) \cdot (x^2 \cdot x) \cdot y.\]
Using the product rule for exponents, where \(x^2 \cdot x = x^2 \cdot x^1 = x^{2+1} = x^3\), we obtain:
\[15 \cdot x^3 \cdot y = 15x^3y.\]
The expression is now simplified with only positive exponents.

  1. Simplifying Expression (b): \((x^2y^3)(x^{-2}y^{-2})\)

To simplify \((x^2y^3)(x^{-2}y^{-2})\), we group the terms with the same base:
\[(x^2 \cdot x^{-2}) \cdot (y^3 \cdot y^{-2}).\]
Using the product rule, \(x^{2+(-2)} = x^0\) and \(y^{3+(-2)} = y^1\).
Since any non-zero variable raised to the power of 0 is 1 (i.e., \(x^0 = 1\)), the expression becomes:
\[1 \cdot y^1 = y.\]
The simplified result is \(y\).

  1. Simplifying Expression (c): \(\frac{x^3}{x^{-2}}\)

To simplify \(\frac{x^3}{x^{-2}}\), we use the quotient rule for exponents:
\[\frac{x^m}{x^n} = x^{m-n}.\]
Applying this with \(m=3\) and \(n=-2\):
\[x^{3 - (-2)} = x^{3+2} = x^5.\]
The simplified result is \(x^5\).

  1. Simplifying Expression (d): \((2x^{-1})^3\)

To simplify \((2x^{-1})^3\), we use the power of a product rule, \((ab)^n = a^n \cdot b^n\):
\[2^3 \cdot (x^{-1})^3.\]
Now, calculate the power of each part: \(2^3 = 8\) and \((x^{-1})^3 = x^{-1 \cdot 3} = x^{-3}\).
This gives us \(8 \cdot x^{-3}\).
Finally, apply the negative exponent rule to express the result with a positive exponent:
\[8 \cdot \frac{1}{x^3} = \frac{8}{x^3}.\]
The simplified result is \(\frac{8}{x^3}\).

Examples

In science and engineering, simplifying expressions with exponents is crucial for managing units and scientific notation. For example, when calculating the physical volume of a space using measurements in centimeters (cm), if a formula results in expressions like \(\frac{cm^3}{cm^{-2}}\), simplifying them allows for easy calculation and correct interpretation. These basic rules are the building blocks for more advanced calculations in physics, chemistry, and computer science.

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