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Section 2.2 Simplifying Radicals
Subsection 2.2.1 Examples
Exercise Group 2.2.1 . Simplifying Square Roots.
Simplify each radical, if possible.
(a) \(\sqrt{8}\)
(b) \(\sqrt{12}\)
(c) \(\sqrt{32}\)
(d) \(\sqrt{50}\)
(e) \(\sqrt{18}\)
(f) \(\sqrt{27}\)
(g) \(\sqrt{48}\)
(h) \(\sqrt{33}\) Answer .
\(\sqrt{33}\) (already simplest)
(i) \(\sqrt{75}\)
(j) \(\sqrt{90}\)
(k) \(\sqrt{108}\)
(l) \(\sqrt{600}\)
(m) \(\sqrt{54}\)
(n) \(\sqrt{28}\)
(o) \(\sqrt{73}\) Answer .
\(\sqrt{73}\) (already simplest)
(p) \(\sqrt{112}\)
(q) \(\sqrt{24}\)
(r) \(\sqrt{80}\)
(s) \(\sqrt{98}\)
(t) \(\sqrt{56}\)
(u) \(\sqrt{91}\) Answer .
\(\sqrt{91}\) (already simplest)
(v) \(\sqrt{200}\)
(w) \(\sqrt{392}\)
(x) \(\sqrt{1575}\)
Exercise Group 2.2.2 . Simplifying Higher Roots.
Simplify each radical, if possible.
(a) \(\sqrt[3]{16}\)
(b) \(\sqrt[3]{81}\)
(c) \(\sqrt[3]{135}\)
(d) \(\sqrt[3]{24}\)
(e) \(\sqrt[3]{256}\)
(f) \(\sqrt[3]{-128}\)
(g) \(\sqrt[3]{60}\) Answer .
\(\sqrt[3]{60}\) (already simplest)
(h) \(\sqrt[3]{24}\)
(i) \(\sqrt[3]{192}\)
(j) \(\sqrt[4]{32}\)
(k) \(\sqrt[3]{40}\)
(l) \(\sqrt[4]{32}\)
Exercise Group 2.2.3 . More Radical Simplification.
Simplify each radical, if possible.
(a) \(\sqrt[3]{375}\)
(b) \(\sqrt[4]{48}\)
(c) \(\sqrt[3]{54}\)
(d) \(\sqrt[3]{243}\)
(e) \(\sqrt[3]{40}\)
(f) \(\sqrt[4]{405}\)
(g) \(\sqrt[3]{-432}\)
(h) \(\sqrt[4]{1250}\)
(i) \(\sqrt[3]{100}\) Answer .
\(\sqrt[3]{100}\) (already simplest)
(j) \(\sqrt[4]{176}\)
(k) \(\sqrt[3]{144}\)
(l) \(\sqrt[3]{-500}\)
(m) \(\sqrt[3]{108}\)
(n) \(\sqrt[4]{162}\)
(o) \(\sqrt[4]{128}\)
(p) \(\sqrt[4]{243}\)
Subsection 2.2.2 Condensing Mixed Radicals, Writing Mixed Radicals as Entire Radicals
Mixed radicals can also be condensed into entire radicals, the reverse process of simplifying radicals.
Any number can be written as the square root of its square. For example,
\(2 = \sqrt{2^2}\text{,}\) \(3 = \sqrt{3^2}\text{,}\) etc. This property, along with the multiplication property, can be used to condense mixed radicals into entire radicals.
Exercise Group 2.2.4 . Mixed to Entire Radicals (Square Roots).
Write each mixed radical as an entire radical.
(a) \(5\sqrt{2}\)
(b) \(6\sqrt{2}\)
(c) \(7\sqrt{2}\)
(d) \(8\sqrt{2}\)
(e) \(5\sqrt{3}\)
(f) \(6\sqrt{3}\)
(g) \(7\sqrt{3}\)
(h) \(4\sqrt{3}\)
(i) \(8\sqrt{3}\)
(j) \(3\sqrt{2}\)
(k) \(4\sqrt{2}\)
(l) \(6\sqrt{5}\)
(m) \(5\sqrt{6}\)
(n) \(7\sqrt{7}\)
(o) \(15\sqrt{6}\)
Similarly, any number can be written as the cube root of its cube, or in general the
\(n\) th root of its
\(n\) th power.
Exercise Group 2.2.5 . Mixed to Entire Radicals (Higher Roots).
Write each mixed radical as an entire radical.
(a) \(2\sqrt[3]{2}\)
(b) \(3\sqrt[3]{3}\)
(c) \(4\sqrt[3]{3}\)
(d) \(2\sqrt[5]{2}\)
(e) \(5\sqrt[3]{2}\)
(f) \(2\sqrt[3]{9}\)
(g) \(3\sqrt[5]{4}\)
(h) \(2\sqrt[5]{3}\)
(i) \(3\sqrt[3]{4}\)
(j) \(2\sqrt[3]{7}\)
(k) \(6\sqrt[4]{3}\)
(l) \(2\sqrt[4]{5}\)
(m) \(7\sqrt[4]{2}\)
(n) \(4\sqrt[5]{3}\)
(o) \(3\sqrt[3]{2}\)
(p) \(2\sqrt[3]{4}\)
(q) \(10\sqrt[3]{5}\)
(r) \(4\sqrt[4]{2}\)
