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Magazine Chapter 10: Problem 126
Simplify. $$\sqrt{27 a^{5}(b+1)} \sqrt[3]{81 a(b+1)^{4}}$$
Short Answer
Expert verified
9a^{2} (b+1)^{11/6} \sqrt[3]{3a}
Step by step solution
01
Simplify the square root
First, simplify the square root expression \(\sqrt{27 a^{5}(b+1)}\) by breaking it into products of simpler expressions. \[ 27 a^{5}(b+1) = 3^{3} a^{5}(b+1) \] Therefore, \(\sqrt{27 a^{5}(b+1)} = \sqrt{3^{3} a^{5}(b+1)} = 3 a^{2}\sqrt{a(b+1)}\).
02
Simplify the cube root
Next, simplify the cube root expression \(\sqrt[3]{81 a(b+1)^{4}}\). Note that \[ 81 a(b+1)^{4} = 3^{4} a(b+1)^{4} \] Therefore, \(\sqrt[3]{81 a(b+1)^{4}} = \sqrt[3]{3^{4} a(b+1)^{4}} = 3 \sqrt[3]{3} (b+1)\sqrt[3]{b+1}\).
03
Multiply the simplified expressions
Now, combine the simplified forms: \[ 3 a^{2}\sqrt{a(b+1)} \times 3 (b+1)\sqrt[3]{3}(b+1) \] Multiply the numerical coefficients first: \[ 3 \times 3 = 9 \] Therefore, \(9 a^{2} (b+1) \sqrt{a(b+1)} \sqrt[3]{3(b+1)}\).
04
Combine the exponential terms
Combine the exponents and roots: \(9a^{2}(b+1) \sqrt{a} \sqrt{b+1} \sqrt[3]{3(b+1)}\) can be further simplified by combining \(\sqrt{b+1}\) and \(\sqrt[3]{b+1}\): \[ \sqrt{b+1} \times \sqrt[3]{b+1} = (b+1)^{1/2 + 1/3} = (b+1)^{5/6} \] Therefore, \(9a^{2}(b+1)^{11/6} \sqrt[3]{3a}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponents
Exponents are a shorthand way to represent repeated multiplication of the same number by itself. For example,聽\( a^5 \)聽means聽\( a \times a \times a \times a \times a \). This is useful for simplifying expressions involving large numbers or variables raised to high powers.
The laws of exponents make it easier to work with these expressions. A few important rules are:
The laws of exponents make it easier to work with these expressions. A few important rules are:
- Product Rule:聽\( a^m \times a^n = a^{m+n} \)
- Quotient Rule:聽\( \frac{a^m}{a^n} = a^{m-n} \)
- Power of a Power Rule:聽\( (a^m)^n = a^{mn} \)
Square Root
The square root of a number聽\( x \)聽is a value that, when multiplied by itself, gives the number聽\( x \). It is denoted as聽\( \sqrt{x} \). For example,聽\( \sqrt{16} = 4 \), because聽\( 4 \times 4 = 16 \). It's also known that the square root of a product can be broken down into the product of square roots:
- \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \)
Cube Root
The cube root of a number聽\( x \)聽is a value that, when used three times in multiplication, gives the number聽\( x \). It's represented as聽\( \sqrt[3]{x} \). For example,聽\( \sqrt[3]{27} = 3 \), because聽\( 3 \times 3 \times 3 = 27 \). Similar to square roots, cube roots also follow a property that is helpful for simplification:
- \( \sqrt[3]{ab} = \sqrt[3]{a} \times \sqrt[3]{b} \)
Multiplication of Radicals
Radicals can be multiplied together by multiplying the numbers inside the radical signs, provided they have the same root index. This follows the rule:
- \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \)
- \( 3 \times 3 = 9 \)
