44. Surds and indices problems

Surds and indices problems involve manipulating expressions with exponents (indices) and irrational roots (surds). Indices deal with powers and their rules (e.g.,

a^m \cdot a^n = a^{m+n}

), while surds involve simplifying or rationalizing expressions with roots (e.g.,

\sqrt{2}

\sqrt[3]{5}

). These problems often require applying exponent laws, simplifying radicals, or solving equations involving powers and roots.

Below are 10 surds and indices problems with step-by-step solutions and detailed explanations. Each problem includes the problem statement, solution procedure, verification, and an explanation of the concept. The problems are diverse, covering exponent laws, surd simplification, rationalizing denominators, and solving equations, ensuring they are distinct from previous problem types (e.g., sequences or age problems). They focus on mathematical expressions and avoid unrelated contexts (e.g., alcohol or mixing solutions).

Problem 1

Simplify the expression:

2^3 \cdot 2^5

Step-by-Step Solution:

Apply the product rule for indices: For the same base,
a^m \cdot a^n = a^{m+n}
.
2^3 \cdot 2^5 = 2^{3+5} = 2^8
Calculate the value:
2^8 = 256
Verify: Compute separately:
2^3 = 8
,
2^5 = 32
. Then,
8 \cdot 32 = 256
, which matches
2^8
.
Answer: The simplified expression is
2^8 = 256
.

Explanation: The product rule combines exponents when multiplying powers of the same base. This is a fundamental law of indices, ensuring efficient simplification of exponential expressions. The verification confirms the rule by computing the product directly.

Problem 2

Simplify the expression:

\frac{5^7}{5^4}

Step-by-Step Solution:

Apply the quotient rule for indices: For the same base,
\frac{a^m}{a^n} = a^{m-n}
.
\frac{5^7}{5^4} = 5^{7-4} = 5^3
Calculate the value:
5^3 = 125
Verify: Compute separately:
5^7 = 78,125
,
5^4 = 625
. Then,
\frac{78,125}{625} = 125
, which matches
5^3
.
Answer: The simplified expression is
5^3 = 125
.

Explanation: The quotient rule subtracts the exponent of the denominator from the numerator’s exponent for the same base. This simplifies division of powers efficiently. The verification ensures accuracy by performing the division directly.

Problem 3

Simplify the expression:

(3^2)^4

Step-by-Step Solution:

Apply the power rule for indices: For a power raised to another power,
(a^m)^n = a^{m \cdot n}
.
(3^2)^4 = 3^{2 \cdot 4} = 3^8
Calculate the value:
3^8 = 6,561
Verify: Compute step-by-step:
3^2 = 9
, then
9^4 = (9^2)^2 = 81^2 = 6,561
, which matches
3^8
.
Answer: The simplified expression is
3^8 = 6,561
.

Explanation: The power rule multiplies the exponents when a power is raised to another power. This allows simplification of nested exponents. The verification confirms the result by computing the expression in an alternative way.

Problem 4

Simplify the surd:

\sqrt{72}

Step-by-Step Solution:

Factorize the number: Break 72 into its prime factors or identifiable squares:
72 = 36 \cdot 2 = 6^2 \cdot 2
Simplify the square root: Use the property
\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}
for
a, b \geq 0
:
\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6 \sqrt{2}
Verify: Check if
6 \sqrt{2}
is correct:
(6 \sqrt{2})^2 = 6^2 \cdot (\sqrt{2})^2 = 36 \cdot 2 = 72
The expression is simplified, as 2 has no perfect square factors.
Answer: The simplified form is
6 \sqrt{2}
.

Explanation: Simplifying a surd involves factoring out perfect squares from the radicand and taking their square roots. Here, 36 is a perfect square, leaving

\sqrt{2}

as the irrational component. This process reduces the surd to its simplest form.

Problem 5

Rationalize the denominator:

\frac{3}{\sqrt{5}}

Step-by-Step Solution:

Identify the denominator: The denominator is
\sqrt{5}
, a surd.
Multiply numerator and denominator by the surd: To eliminate the surd, multiply by
\sqrt{5}
:
\frac{3}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{3 \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} = \frac{3 \sqrt{5}}{5}
Verify: The denominator is now 5, a rational number. Check equivalence:
\frac{3}{\sqrt{5}} \approx \frac{3}{2.236} \approx 1.341, \quad \frac{3 \sqrt{5}}{5} \approx \frac{3 \cdot 2.236}{5} \approx 1.341
The expressions are equivalent.
Answer: The rationalized form is
\frac{3 \sqrt{5}}{5}
.

Explanation: Rationalizing the denominator removes the surd by multiplying by a form that makes the denominator rational (here,

\sqrt{5} \cdot \sqrt{5} = 5

). This is a standard technique for simplifying fractions with irrational denominators.

Problem 6

Solve the equation:

2^x = 16

Step-by-Step Solution:

Express both sides with the same base: Recognize that 16 is a power of 2:
16 = 2^4
So, the equation becomes:
2^x = 2^4
Equate the exponents: Since the bases are the same, the exponents must be equal:
x = 4
Verify: Substitute
x = 4
:
2^4 = 16
The equation holds.
Answer:
x = 4
.

Explanation: This problem uses the property that if

a^x = a^y

, then

x = y

for the same base ( a ). Expressing 16 as

2^4

allows direct comparison of exponents, a common method for solving exponential equations.

Problem 7

Simplify the expression:

\sqrt{50} + \sqrt{18} - \sqrt{8}

Step-by-Step Solution:

Simplify each surd:
- \sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5 \sqrt{2}
- \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3 \sqrt{2}
- \sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2 \sqrt{2}
Combine like terms: The expression becomes:
5 \sqrt{2} + 3 \sqrt{2} - 2 \sqrt{2}
Combine coefficients of
\sqrt{2}
:
(5 + 3 - 2) \sqrt{2} = 6 \sqrt{2}
Verify: Check numerically (approximate):
\sqrt{50} \approx 7.071, \quad \sqrt{18} \approx 4.243, \quad \sqrt{8} \approx 2.828
7.071 + 4.243 - 2.828 \approx 8.486
6 \sqrt{2} \approx 6 \cdot 1.414 \approx 8.484
The values are close, confirming the simplification.
Answer: The simplified form is
6 \sqrt{2}
.

Explanation: Simplifying surds involves reducing each to its simplest form and combining like terms (terms with the same radical). Factoring out perfect squares and treating surds like variables allows algebraic combination, a key technique in surd manipulation.

Problem 8

Solve the equation:

x^2 = 25

Step-by-Step Solution:

Isolate the variable: The equation is:
x^2 = 25
Take the square root of both sides: Since
x^2 = 25
, apply the square root:
x = \pm \sqrt{25} = \pm 5
Verify:
- For
  x = 5
  :
  5^2 = 25
  , satisfies.
- For
  x = -5
  :
  (-5)^2 = 25
  , satisfies.
Answer:
x = 5
or
x = -5
.

Explanation: This is a simple equation involving indices (exponent 2). Taking the square root accounts for both positive and negative solutions, as squaring eliminates the sign. This problem introduces solving equations with even exponents, which typically yield two solutions.

Problem 9

Simplify the expression:

\frac{2^3 \cdot 3^2}{2^5 \cdot 3^4}

Step-by-Step Solution:

Separate the bases: Rewrite the expression:
\frac{2^3 \cdot 3^2}{2^5 \cdot 3^4} = \frac{2^3}{2^5} \cdot \frac{3^2}{3^4}
Apply the quotient rule for each base:
- For base 2:
  \frac{2^3}{2^5} = 2^{3-5} = 2^{-2}
- For base 3:
  \frac{3^2}{3^4} = 3^{2-4} = 3^{-2}
Combine the terms:
2^{-2} \cdot 3^{-2} = (2 \cdot 3)^{-2} = 6^{-2}
Simplify the negative exponent:
6^{-2} = \frac{1}{6^2} = \frac{1}{36}
Verify: Compute separately:
- Numerator:
  2^3 = 8
  ,
  3^2 = 9
  , so
  8 \cdot 9 = 72
- Denominator:
  2^5 = 32
  ,
  3^4 = 81
  , so
  32 \cdot 81 = 2,592
- Fraction:
  \frac{72}{2,592} = \frac{72 \div 72}{2,592 \div 72} = \frac{1}{36}
  , matches.
Answer: The simplified form is
\frac{1}{36}
.

Explanation: This problem combines quotient and product rules for indices across multiple bases. Negative exponents arise when the denominator’s exponent is larger, and combining bases allows further simplification. The verification ensures accuracy by computing the original fraction.

Problem 10

Rationalize the denominator:

\frac{4}{2 + \sqrt{3}}

Step-by-Step Solution:

Identify the denominator: The denominator is
2 + \sqrt{3}
, a binomial with a surd.
Use the conjugate: Multiply numerator and denominator by the conjugate of the denominator,
2 - \sqrt{3}
, to rationalize:
\frac{4}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{4 (2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})}
Simplify the denominator: Use the difference of squares:
(2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1
Simplify the numerator:
4 (2 - \sqrt{3}) = 8 - 4 \sqrt{3}
Combine:
\frac{8 - 4 \sqrt{3}}{1} = 8 - 4 \sqrt{3}
Verify: Check numerically:
2 + \sqrt{3} \approx 2 + 1.732 = 3.732, \quad \frac{4}{3.732} \approx 1.072
8 - 4 \sqrt{3} \approx 8 - 4 \cdot 1.732 \approx 8 - 6.928 \approx 1.072
The values match, confirming the result.
Answer: The rationalized form is
8 - 4 \sqrt{3}
.

Explanation: Rationalizing a binomial denominator involves multiplying by the conjugate to eliminate the surd, using the difference of squares property. This transforms the denominator into a rational number, simplifying the expression. The verification confirms equivalence through numerical approximation.

General Notes on Surds and Indices Problems

Key Concepts:
- Indices: Exponents follow laws like:
  - Product rule:
    a^m \cdot a^n = a^{m+n}
  - Quotient rule:
    \frac{a^m}{a^n} = a^{m-n}
  - Power rule:
    (a^m)^n = a^{m \cdot n}
  - Negative exponent:
    a^{-n} = \frac{1}{a^n}
  - Zero exponent:
    a^0 = 1
    (for
    a \neq 0
    )
- Surds: Irrational roots (e.g.,
  \sqrt{2}
  ) are simplified by factoring out perfect squares or rationalizing denominators.
  - Simplification:
    \sqrt{ab} = \sqrt{a} \cdot \sqrt{b}
    , factor out perfect squares.
  - Rationalization: Multiply by the surd or its conjugate to make the denominator rational.
Key Steps:
1. Identify the operation (e.g., multiplication, division, or root simplification).
2. Apply relevant indices laws or surd properties (e.g., factorize for surds, combine exponents for indices).
3. Simplify the expression or solve the equation, ensuring all steps follow mathematical rules.
4. Verify by computing numerically, checking equivalence, or substituting solutions.
Challenges:
- Ensure bases are the same when applying indices rules.
- Avoid errors in factoring surds or handling negative exponents.
- For binomial denominators, use the conjugate correctly to rationalize.
- Solving equations may require expressing numbers as powers or taking roots, considering all possible solutions (e.g., positive and negative for even roots).
Applications: These problems appear in algebra, calculus, physics, and engineering, where exponents model growth, decay, or scaling, and surds arise in geometric or trigonometric calculations.
Verification: Verify simplifications by computing original and simplified expressions numerically or algebraically. For equations, substitute solutions back to ensure they satisfy the original equation.

These problems cover a range of surds and indices concepts, from basic exponent laws to advanced surd rationalization and equation solving, demonstrating their application in algebraic manipulation.

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44. Surds and indices problems

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