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Exercises Expressions, exponents and radicals

Example 7 from Algebra and Trigonometry, 1.1 Real Numbers: Algebra Essentials

Using Properties of Real Numbers

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

ⓐ $3 \cdot 6 + 3 \cdot 4$
ⓑ $(5 + 8) + (−8)$
ⓒ $6 - (15 + 9)$
ⓓ $\frac{4}{7} \cdot (\frac{2}{3} \cdot \frac{7}{4})$
ⓔ $100 \cdot [0.75 + (−2.38)]$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
1. ⓐ
  $\begin{matrix} 3 \cdot 6 + 3 \cdot 4 & = & 3 \cdot (6 + 4) & Distributive property. \\ = & 3 \cdot 10 & Simplify. \\ = & 30 & Simplify. \end{matrix}$
2. ⓑ
  $\begin{matrix} (5 + 8) + (−8) & = & 5 + [8 + (−8)] & Associative property of addition. \\ = & 5 + 0 & Inverse property of addition. \\ = & 5 & Identity property of addition. \end{matrix}$
3. ⓒ
  $\begin{matrix} 6 - (15 + 9) & = & 6 + [(−15) + (−9)] & Distributive property. \\ = & 6 + (−24) & Simplify. \\ = & −18 & Simplify. \end{matrix}$
4. ⓓ
  $\begin{matrix} \frac{4}{7} \cdot (\frac{2}{3} \cdot \frac{7}{4}) & = & \frac{4}{7} \cdot (\frac{7}{4} \cdot \frac{2}{3}) & Commutative property of multiplication. \\ = & (\frac{4}{7} \cdot \frac{7}{4}) \cdot \frac{2}{3} & Associative property of multiplication. \\ = & 1 \cdot \frac{2}{3} & Inverse property of multiplication. \\ = & \frac{2}{3} & Identity property of multiplication. \end{matrix}$
5. ⓔ
  $\begin{matrix} 100 \cdot [0.75 + (- 2.38)] & = & 100 \cdot 0.75 + 100 \cdot (−2.38) & Distributive property. \\ = & 75 + (−238) & Simplify. \\ = & −163 & Simplify. \end{matrix}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

ⓐ $(- \frac{23}{5}) \cdot [11 \cdot (- \frac{5}{23})]$
ⓑ $5 \cdot (6.2 + 0.4)$
ⓒ $18 - (7 −15)$
ⓓ $\frac{17}{18} + [\frac{4}{9} + (- \frac{17}{18})]$
ⓔ $6 \cdot (−3) + 6 \cdot 3$

How did it go?

PreviousUsing the Order of Operations Next Describing Algebraic Expressions

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 1.1, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.