So, you were trying to it is in a an excellent test taker and also practice because that the GRE with PowerPrep online. Buuuut climate you had actually some questions about the quant section—specifically concern 12 that the second Quantitative ar of practice Test 1. Those questions experimentation our knowledge of **Exponents and also Roots** deserve to be sort of tricky, but never fear, chrischona2015.org has acquired your back!

## Survey the Question

Let’s find the difficulty for clues as to what it will be testing, as this will help transition our minds come think about what kind of math knowledge we’ll usage to settle this question. Pay attention to any kind of words the sound math-specific and anything special around what the number look like, and mark them on ours paper.

You are watching: What is the least integer n such that

We see that $n$ is an **exponent** in our question, therefore we’ll most likely use our **Exponents and Roots** mathematics skill. And since we have actually an **inequality** in our question, we’ll most likely use our **Solving linear Inequalities** mathematics skill. Let’s keep what we’ve learned around these an abilities at the reminder of our minds as we technique this question.

## What carry out We Know?

Let’s very closely read with the question and make a perform of the points that us know.

We desire to find the least integer worth of $n$We recognize that: $1/2^n## Develop a Plan

We want to find the the very least value the $n$, so let’s try to leveling the inequality in our concern until it’s much easier to see what worths $n$ can have.

## Solve the Question

When addressing an equation or inequality because that a variable, it’s usually best to rearrange the equation until we have that variable in the numerator of any fractions, diverted by itself. Looking in ~ our inequality, let’s begin by multiply both political parties of the inequality by $2^n$ to eliminate it from the denominator of the fraction. **Anytime we multiply or division an inequality by a number, we need to dual check to make certain that number is positive. If the is negative, we have to reverse the inequality’s sign direction**. We know that $2$ elevated to any kind of exponent will offer us a positive result, so we don’t have to worry around reversing the inequality’s authorize direction here.

$1/2^n$ | $ |

Let’s proceed isolating the $n$ by separating both sides of the inequality by $0.001$.

$1$ | $ |

Ah, this is much easier to understand! therefore $2^n$ should be greater than $1,000$. Although there technically is a mathematical means to solve for $n$, that is definitely past the scope of the GRE. Therefore instead, let’s usage our scrap paper to make a table of different $n$ and $2^n$ values, together with whether or no $2^n$ is greater than $1,000$. For this reason let’s start with $n=1$ and use ours math knowledge or GRE calculator to keep multiplying by $2$.

$;;;;;;;;;;;;;;;n;;;;;;;;;;;;;;;$ | $2^n$ | Is $2^n$ better than $1,000$? |

$1$ | $2$ | No |

$2$ | $4$ | No |

$3$ | $8$ | No |

$4$ | $16$ | No |

$5$ | $32$ | No |

$6$ | $64$ | No |

$7$ | $128$ | No |

$8$ | $256$ | No |

$9$ | $512$ | No |

$10$ | $1,024$ | Yes |

Nicely done! so the looks choose $n=10$ is the very first integer value for $n$ where $2^n$ is better than $1,000$. **The correct answer is A, $10$**.

## What Did us Learn

Simplifying the inequality to be definitely very helpful. It readjusted a very abstract spring inequality and also turned it into an much easier to know concept: detect a worth for $n$ whereby $2^n$ would be better than $1,000$. So a great lesson to find out is **simplify anytime possible**.

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