Podcast
Questions and Answers
When simplifying a nested radical expression like $\sqrt{a\sqrt{a\sqrt{a}}}$, which property of exponents is most crucial for combining the terms?
When simplifying a nested radical expression like $\sqrt{a\sqrt{a\sqrt{a}}}$, which property of exponents is most crucial for combining the terms?
- The quotient of powers rule: $\frac{a^m}{a^n} = a^{m-n}$
- The power of a power rule: $(a^m)^n = a^{mn}$
- The product of powers rule: $a^m \cdot a^n = a^{m+n}$ (correct)
- The power of a product rule: $(ab)^n = a^n b^n$
What distinguishes the 'Shelter Approach' from the 'Hair Approach' in simplifying radical expressions?
What distinguishes the 'Shelter Approach' from the 'Hair Approach' in simplifying radical expressions?
- The 'Shelter Approach' is used for expressions involving even roots, while the 'Hair Approach' is for odd roots.
- The 'Shelter Approach' applies only to real numbers, while the 'Hair Approach' applies to complex numbers.
- The 'Shelter Approach' simplifies radicals by rationalizing denominators, while the 'Hair Approach' simplifies by multiplying by a conjugate.
- The 'Shelter Approach' deals with finite radical expressions, while the 'Hair Approach' deals with infinite radical expressions. (correct)
In the expression $x = \sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}}$, what algebraic technique allows us to rewrite the infinite nested radical as a simple equation?
In the expression $x = \sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}}$, what algebraic technique allows us to rewrite the infinite nested radical as a simple equation?
- Factoring out a common term from inside the square root
- Substituting a variable for a finite portion of the expression.
- Multiplying both sides by the conjugate of the expression inside the square root.
- Squaring both sides and recognizing the repeating pattern (correct)
When solving for $x$ in the equation $x = \sqrt{1 + x}$, which of the following steps is crucial to ensure the validity of the solution?
When solving for $x$ in the equation $x = \sqrt{1 + x}$, which of the following steps is crucial to ensure the validity of the solution?
Given the equation $2 = \sqrt[4]{8\sqrt[3]{2\sqrt{8x}}}$, what is the most efficient first step to isolate $x$?
Given the equation $2 = \sqrt[4]{8\sqrt[3]{2\sqrt{8x}}}$, what is the most efficient first step to isolate $x$?
The quadratic formula is used to solve equations of the form $ax^2 + bx + c = 0$. For the equation derived from the infinite radical $x = \sqrt{1 + x}$, which coefficients $a$, $b$, and $c$ should be used in the quadratic formula?
The quadratic formula is used to solve equations of the form $ax^2 + bx + c = 0$. For the equation derived from the infinite radical $x = \sqrt{1 + x}$, which coefficients $a$, $b$, and $c$ should be used in the quadratic formula?
If $x = a^{7/8}$, which of the following expressions correctly represents $x$ in radical form?
If $x = a^{7/8}$, which of the following expressions correctly represents $x$ in radical form?
In simplifying nested radicals, why is it important to express all terms with a common base before applying exponent rules?
In simplifying nested radicals, why is it important to express all terms with a common base before applying exponent rules?
When dealing with infinite radical expressions, what condition must be met for the expression to converge to a finite value?
When dealing with infinite radical expressions, what condition must be met for the expression to converge to a finite value?
Given the solution to the infinite radical expression $x = \sqrt{1 + x}$, which of the two solutions obtained from the quadratic formula is valid in the context of real numbers and why?
Given the solution to the infinite radical expression $x = \sqrt{1 + x}$, which of the two solutions obtained from the quadratic formula is valid in the context of real numbers and why?
Flashcards
Shelter Approach
Shelter Approach
A method used to simplify radical expressions with a finite number of terms.
Hair Approach
Hair Approach
A method used to simplify radical expressions with an infinite number of terms.
Adding exponents
Adding exponents
To simplify expressions with the same variable, add the exponents.
Finite radical expression.
Finite radical expression.
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Infinite radical expression
Infinite radical expression
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Golden Ratio
Golden Ratio
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Study Notes
- The notes cover the topic of exponents and radicals in algebra
- The date is 06/11/2019
- There are two solution approaches:
- Shelter approach: For finite radical expressions
- Hair approach: For infinite radical expressions
Problem #1
- Find the value of x
- The equation is x = √(a√a√a)
- Rewrite the equation: x = {a[a(a)^1/2]^1/2}^1/2
- Expressed with exponents: x = a * [a(a^(1/2))]^(1/4) = a^(1/2) * a^(1/4) * a^(1/8)
- Same variables, add exponents
- Simplify exponents: x = a^(1/2 + 1/4 + 1/8)
- Therefore x = a^(7/8)
Problem #2 (Civil Engineering Exam 1991)
- Solve for x: 2 = fourth root of (cube root of (8 * square root of 2 * square root of 8x)))
- Rewrite the equation: 2 = [(8)^(1/4) * (2)^(1/12) * (8x)^(1/24)]^24
- Simplify: (2)^24 = (8)^6 * (2)^2 * (8x)^1
- Therefore x = 2
Problem #3
- Solve for x: x = √(1 + √(1 + √(1 + ...)))
- Rewrite the equation: x = √(1 + x)
- To remove the square root, square both sides
- New equation: x^2 = 1 + x
- Rearrange to quadratic equation: x^2 -x -1 = 0
- Solutions for x:
- x1 = 1.618
- x2 = -0.618
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