Radicals & Absolute Values - PSAT Math
Card 1 of 30
What is the value of $|3-8|$?
What is the value of $|3-8|$?
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$5$. $3 - 8 = -5$, and $|-5| = 5$.
$5$. $3 - 8 = -5$, and $|-5| = 5$.
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What is the solution set of $|x|\le 3$ written as an interval?
What is the solution set of $|x|\le 3$ written as an interval?
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$[-3,3]$. Distance from 0 at most 3 includes the endpoints.
$[-3,3]$. Distance from 0 at most 3 includes the endpoints.
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What is the value of $|x-4|$ when $x=1$?
What is the value of $|x-4|$ when $x=1$?
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$3$. $1 - 4 = -3$, and $|-3| = 3$.
$3$. $1 - 4 = -3$, and $|-3| = 3$.
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What is the solution set of $|x-3|=5$?
What is the solution set of $|x-3|=5$?
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$x=8$ or $x=-2$. $x - 3 = 5$ or $x - 3 = -5$, giving two solutions.
$x=8$ or $x=-2$. $x - 3 = 5$ or $x - 3 = -5$, giving two solutions.
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What is the solution set of $|2x|=10$?
What is the solution set of $|2x|=10$?
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$x=5$ or $x=-5$. $2x = 10$ or $2x = -10$, so $x = 5$ or $x = -5$.
$x=5$ or $x=-5$. $2x = 10$ or $2x = -10$, so $x = 5$ or $x = -5$.
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What is the solution set of $|x|<4$ written as an interval?
What is the solution set of $|x|<4$ written as an interval?
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$(-4,4)$. Distance from 0 less than 4 means between -4 and 4.
$(-4,4)$. Distance from 0 less than 4 means between -4 and 4.
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What is $\sqrt{a^2}$ in terms of $|a|$ for real $a$?
What is $\sqrt{a^2}$ in terms of $|a|$ for real $a$?
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$\sqrt{a^2}=|a|$. Square root of a square always gives the absolute value.
$\sqrt{a^2}=|a|$. Square root of a square always gives the absolute value.
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What is $\sqrt{x^2}$ when $x<0$?
What is $\sqrt{x^2}$ when $x<0$?
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$-x$. When $x < 0$, $|x| = -x$, and $\sqrt{x^2} = |x|$.
$-x$. When $x < 0$, $|x| = -x$, and $\sqrt{x^2} = |x|$.
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What is the value of $|{-12}|$?
What is the value of $|{-12}|$?
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$12$. Absolute value makes negative numbers positive.
$12$. Absolute value makes negative numbers positive.
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What is the value of $\sqrt{72}$ in simplest radical form?
What is the value of $\sqrt{72}$ in simplest radical form?
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$6\sqrt{2}$. $72 = 36 \times 2 = 6^2 \times 2$, so factor out the perfect square.
$6\sqrt{2}$. $72 = 36 \times 2 = 6^2 \times 2$, so factor out the perfect square.
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What is the value of $\sqrt{50}$ in simplest radical form?
What is the value of $\sqrt{50}$ in simplest radical form?
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$5\sqrt{2}$. $50 = 25 \times 2 = 5^2 \times 2$, so factor out the perfect square.
$5\sqrt{2}$. $50 = 25 \times 2 = 5^2 \times 2$, so factor out the perfect square.
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What is the value of $\frac{\sqrt{18}}{\sqrt{2}}$ in simplest form?
What is the value of $\frac{\sqrt{18}}{\sqrt{2}}$ in simplest form?
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$3$. $\frac{\sqrt{18}}{\sqrt{2}} = \sqrt{\frac{18}{2}} = \sqrt{9} = 3$.
$3$. $\frac{\sqrt{18}}{\sqrt{2}} = \sqrt{\frac{18}{2}} = \sqrt{9} = 3$.
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What is the product rule for radicals: $\sqrt{a},\sqrt{b}$ for $a\ge 0$ and $b\ge 0$?
What is the product rule for radicals: $\sqrt{a},\sqrt{b}$ for $a\ge 0$ and $b\ge 0$?
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$\sqrt{a},\sqrt{b}=\sqrt{ab}$. Radicals can be multiplied under one radical sign.
$\sqrt{a},\sqrt{b}=\sqrt{ab}$. Radicals can be multiplied under one radical sign.
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What is the solution set of $|x+1|>3$ written as a union of intervals?
What is the solution set of $|x+1|>3$ written as a union of intervals?
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$(-\infty,-4)\cup(2,\infty)$. $x + 1 < -3$ or $x + 1 > 3$, so $x < -4$ or $x > 2$.
$(-\infty,-4)\cup(2,\infty)$. $x + 1 < -3$ or $x + 1 > 3$, so $x < -4$ or $x > 2$.
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What is the simplified value of $\sqrt{72}$?
What is the simplified value of $\sqrt{72}$?
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$6\sqrt{2}$. $72 = 36 \times 2 = 6^2 \times 2$, so $\sqrt{72} = 6\sqrt{2}$.
$6\sqrt{2}$. $72 = 36 \times 2 = 6^2 \times 2$, so $\sqrt{72} = 6\sqrt{2}$.
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What is the solution set of $|x+1|>2$ written as a union of intervals?
What is the solution set of $|x+1|>2$ written as a union of intervals?
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$(-\infty,-3)\cup(1,\infty)$. $x+1 > 2$ or $x+1 < -2$ gives $x > 1$ or $x < -3$.
$(-\infty,-3)\cup(1,\infty)$. $x+1 > 2$ or $x+1 < -2$ gives $x > 1$ or $x < -3$.
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What is the definition of $\sqrt{a}$ for $a\ge 0$?
What is the definition of $\sqrt{a}$ for $a\ge 0$?
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$\sqrt{a}$ is the nonnegative number $r$ such that $r^2=a$. The square root gives the positive value whose square equals $a$.
$\sqrt{a}$ is the nonnegative number $r$ such that $r^2=a$. The square root gives the positive value whose square equals $a$.
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What is the definition of $|x|$ in terms of cases?
What is the definition of $|x|$ in terms of cases?
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$|x|=x$ if $x\ge 0$; $|x|=-x$ if $x<0$. Absolute value keeps positive values unchanged, makes negative values positive.
$|x|=x$ if $x\ge 0$; $|x|=-x$ if $x<0$. Absolute value keeps positive values unchanged, makes negative values positive.
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What is the principal value of $\sqrt{81}$?
What is the principal value of $\sqrt{81}$?
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$9$. Since $9^2 = 81$, the principal square root is $9$.
$9$. Since $9^2 = 81$, the principal square root is $9$.
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What is the value of $\sqrt{0.0009}$?
What is the value of $\sqrt{0.0009}$?
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$0.03$. Since $0.03^2 = 0.0009$, the square root is 0.03.
$0.03$. Since $0.03^2 = 0.0009$, the square root is 0.03.
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What is the solution set of $|x-2|\le 3$ written as an interval?
What is the solution set of $|x-2|\le 3$ written as an interval?
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$[-1,5]$. $-3 \le x-2 \le 3$ gives $-1 \le x \le 5$.
$[-1,5]$. $-3 \le x-2 \le 3$ gives $-1 \le x \le 5$.
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What is the value of $\sqrt{\frac{1}{16}}$?
What is the value of $\sqrt{\frac{1}{16}}$?
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$\frac{1}{4}$. Since $(\frac{1}{4})^2 = \frac{1}{16}$, this is the square root.
$\frac{1}{4}$. Since $(\frac{1}{4})^2 = \frac{1}{16}$, this is the square root.
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What is the value of $\sqrt{49}$?
What is the value of $\sqrt{49}$?
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$7$. Since $7^2 = 49$, the square root of 49 is 7.
$7$. Since $7^2 = 49$, the square root of 49 is 7.
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What is the definition of $|x|$ in terms of $x$ and $-x$?
What is the definition of $|x|$ in terms of $x$ and $-x$?
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$|x|$ is the distance from $0$; equals $x$ if $x \ge 0$, else $-x$. Absolute value removes the sign, making negatives positive.
$|x|$ is the distance from $0$; equals $x$ if $x \ge 0$, else $-x$. Absolute value removes the sign, making negatives positive.
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What is the definition of $\sqrt{a}$ for $a \ge 0$?
What is the definition of $\sqrt{a}$ for $a \ge 0$?
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$\sqrt{a}$ is the nonnegative number whose square is $a$. The square root gives the principal (nonnegative) root only.
$\sqrt{a}$ is the nonnegative number whose square is $a$. The square root gives the principal (nonnegative) root only.
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What is the solution set of $|x-2|\le 6$ written as an interval?
What is the solution set of $|x-2|\le 6$ written as an interval?
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$[-4,8]$. $-6 \le x - 2 \le 6$, so $-4 \le x \le 8$.
$[-4,8]$. $-6 \le x - 2 \le 6$, so $-4 \le x \le 8$.
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What is the value of $\sqrt{0.81}$?
What is the value of $\sqrt{0.81}$?
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$0.9$. Since $0.9^2 = 0.81$, the square root is 0.9.
$0.9$. Since $0.9^2 = 0.81$, the square root is 0.9.
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What is the value of $\sqrt{49}$?
What is the value of $\sqrt{49}$?
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$7$. Since $7^2 = 49$, the principal square root of 49 is 7.
$7$. Since $7^2 = 49$, the principal square root of 49 is 7.
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What is the value of $\sqrt{50}$ in simplest radical form?
What is the value of $\sqrt{50}$ in simplest radical form?
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$5\sqrt{2}$. $50 = 25 \cdot 2 = 5^2 \cdot 2$, so $\sqrt{50} = 5\sqrt{2}$.
$5\sqrt{2}$. $50 = 25 \cdot 2 = 5^2 \cdot 2$, so $\sqrt{50} = 5\sqrt{2}$.
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What is the solution set of $|x+1|\ge^2$ written in interval notation?
What is the solution set of $|x+1|\ge^2$ written in interval notation?
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$(-\infty,-3]\cup[1,\infty)$. $|x + 1| \ge 2$ means $x + 1 \le -2$ or $x + 1 \ge 2$, so $x \le -3$ or $x \ge 1$.
$(-\infty,-3]\cup[1,\infty)$. $|x + 1| \ge 2$ means $x + 1 \le -2$ or $x + 1 \ge 2$, so $x \le -3$ or $x \ge 1$.
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