About Operations with Radical Expressions:
We will often be asked to perform operations with radical expressions. When we have "like radicals", we can add or subtract radicals by leaving the radical part unchanged and performing operations with the numbers that are multiplying the radical. Additionally, we will run into problems that involve multiplying radicals.
Test Objectives
- Demonstrate an understanding of "like radicals"
- Demonstrate the ability to add radical expressions
- Demonstrate the ability to subtract radical expressions
- Demonstrate the ability to multiply radical expressions
#1:
Instructions: Simplify each.
Assume all variables are positive real numbers.
$$a)\hspace{.2em}3\sqrt{54}- 3\sqrt{24}$$
$$b)\hspace{.2em}{-}x\sqrt[3]{24x}+ 3\sqrt[3]{3x^4}$$
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#2:
Instructions: Simplify each.
Assume all variables are positive real numbers.
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$$a)\hspace{.2em}{-}2\sqrt[4]{80x^5}+ 2x\sqrt[4]{405x}+ 2x\sqrt[4]{324}$$
$$b)\hspace{.2em}2xy\sqrt[3]{8xyz}+ \sqrt[3]{x^4y^4z}$$
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#3:
Instructions: Simplify each.
Assume all variables are positive real numbers.
$$a)\hspace{.2em}3\sqrt{15}(4 + \sqrt{5})$$
$$b)\hspace{.2em}2\sqrt{20}\cdot 5 \sqrt{5}$$
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#4:
Instructions: Simplify each.
Assume all variables are positive real numbers.
$$a)\hspace{.2em}(\sqrt{3}- 5)(\sqrt{3}+ 5)$$
$$b)\hspace{.2em}(4\sqrt{3x}+ 4)(4\sqrt{3x}- 4)$$
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#5:
Instructions: Simplify each.
Assume all variables are positive real numbers.
$$a)\hspace{.2em}(5\sqrt{3}+ 2)(2\sqrt{3}+ 3)$$
$$b)\hspace{.2em}(2x + \sqrt[3]{y})(4x + 5\sqrt[3]{y})$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}3\sqrt{6}$$
$$b)\hspace{.2em}x \cdot \sqrt[3]{3x}$$
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#2:
Solutions:
$$a)\hspace{.2em}2x \cdot \sqrt[4]{5x}+ 6x \cdot \sqrt{2}$$
$$b)\hspace{.2em}5xy \cdot \sqrt[3]{xyz}$$
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#3:
Solutions:
$$a)\hspace{.2em}12\sqrt{15}+ 15\sqrt{3}$$
$$b)\hspace{.2em}100$$
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#4:
Solutions:
$$a)\hspace{.2em}{-}22$$
$$b)\hspace{.2em}48x-16$$
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#5:
Solutions:
$$a)\hspace{.2em}36 + 19 \sqrt{3}$$
$$b)\hspace{.2em}8x^2 + 14x \sqrt[3]{y}+ 5\sqrt[3]{y^2}$$
